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Perimeter of a triangle

Here you will learn how to find the perimeter of a triangle, including what the perimeter is, how to calculate it and how to solve perimeter word problems.

Students will first learn how to find perimeter as part of measurement and data in 3rd grade.

What is the perimeter of a triangle?

The perimeter of a triangle is the total distance around the outside of the triangle.

For example,

Perimeter is measured in units, like inches.

Perimeter of a Triangle image 1 US

Measuring the perimeter is like starting at one vertex of a triangle and measuring the total distance around the triangle.

Perimeter of a Triangle image 2 US

The perimeter can also be calculated by adding the length of each side.

7 + 7 + 3 = 17, so the perimeter is 17 inches.

What is the perimeter of a triangle?

Common Core State Standards

How does this relate to 3rd grade math?

  • Grade 3 – Measurement and data (3.MD.D.8) Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

How to find the perimeter of a triangle

In order to calculate the perimeter of a triangle:

Add all the side lengths.

Write the final answer with the correct units.

[FREE] Perimeter Check for Understanding Quiz (Grade 3 to 4)

[FREE] Perimeter Check for Understanding Quiz (Grade 3 to 4)

Use this quiz to check your grade 3 to 4 students’ understanding of perimeter. 10+ questions with answers covering a range of 3rd and 4th grade perimeter topics to identify areas of strength and support!

How to find perimeter examples

Example 1: perimeter of an isosceles triangle.

What is the perimeter of the triangle?

Perimeter of a Triangle image 3 US

To find the perimeter (the total distance around the triangle), add all the side lengths:

9 + 9 + 12 = 30

2 Write the final answer with the correct units.

The side lengths are measured in feet, so the total perimeter is in feet.

The perimeter of the triangle is 30 feet.

Example 2: perimeter of a right triangle

Perimeter of a Triangle image 4 US

12 + 20 + 16 = 48

The side lengths are measured in millimeters, so the total perimeter is in millimeters.

The perimeter of the triangle is 48 millimeters.

Example 3: perimeter of a scalene triangle

Perimeter of a Triangle image 5 US

11 + 17 + 8 = 36

The side lengths are measured in centimeters, so the total perimeter is in centimeters.

The perimeter of the triangle is 36 centimeters.

Example 4: perimeter of equilateral triangle

Perimeter of a Triangle image 6 US

To find the perimeter (the total distance around the triangle), add all the equal side lengths:

16 + 16 + 16 = 48

For the perimeter of an equilateral triangle, since the sides of the triangle are the same (congruent), you can also multiply one side length by 3.

3 \times 16 = 48

The side lengths are measured in meters, so the total perimeter is in meters.

The perimeter of the triangle is 48 meters.

Example 5: triangle perimeter word problem

A triangular street sign has side lengths of 18 inches, 18 inches and 16 inches. What is the perimeter of the sign?

18 + 18 + 16 = 52

The side lengths are measured in inches, so the total perimeter is in inches.

The perimeter of the sign is 52 inches.

Example 6: right scalene triangle

The perimeter of the triangle is 75{~cm}. What is the missing side length?

Perimeter of a Triangle image 7 US

\begin{aligned} & 17 \, + \, 31 \, + \,? = 75 \\\\ & 48 \, + \, ? = 75 \end{aligned}

The two sides together are 48. The third side is missing.

To find the missing side, think about what number added to 48 is 75.

Since 48 + 27 = 75, the missing side length is 27.

The side lengths are measured in centimeters and the total perimeter is in centimeters.

The missing side length of the triangle is 27 centimeters.

Teaching tips for the perimeter of a triangle

  • Choose worksheets that have a variety of question types – varying the types of triangles – equilateral, isosceles and scalene (therefore varying the lengths of the sides), solving for the perimeter or a missing side length, word problems with or without visuals.
  • Give students opportunities to measure and solve problems for the perimeter of triangles in the real world.

Easy mistakes to make

  • Thinking the order of adding the sides matters It doesn’t matter the order in which the sides of the triangle are added, because of the commutative property of addition. As long as all sides are added only once, they can be added in any order.
  • Confusing the formulas or solving strategies for perimeter with area It is easy to confuse these two concepts, especially when first learning them. Remember that perimeter is the measurement of the length around a triangle (one-dimensional), and area is the measurement of the space within a triangle (two-dimensional).
  • Adding uncommon units Side lengths need to be the same units before they can be added. If the sides are shown with different units, convert them to a common unit and then add.

Related perimeter lessons

  • How to find perimeter
  • Perimeter of a square
  • Perimeter of a rectangle

Practice perimeter of a triangle questions

1) What is the perimeter of the triangle?

Perimeter of a Triangle image 8 US

The perimeter (the total distance around the triangle) is the sum of the lengths of the three sides:

9 + 15 + 12 = 36

The sides are measured in feet, so the perimeter is also in feet.

The perimeter is 36 feet.

2) What is the perimeter of triangle ABC?

Perimeter of a Triangle image 9 US

18 + 18 + 42 = 78

The sides are measured in centimeters, so the perimeter is also in centimeters.

The perimeter of triangle ABC is 78{~cm}.

3) Which triangle has a perimeter of 41 units?

Perimeter of a Triangle image 10 US

Since the perimeter of the triangle is 41 units, the length of its sides should add up to 41.

Perimeter of a Triangle image 14 US

17 + 17 + 7 = 41

This triangle has a perimeter of 41 units.

4) Which are the sides of a triangle with a perimeter of 50{~cm}?

three sides of 20{~cm}

A triangle has 3 sides. The perimeter of a triangle is the sum of the 3 side lengths:

15 + 20 + 15 = 50

The sides of a triangle 15{~cm}, 20{~cm}, and 15{~cm} have a perimeter of 50{~cm}.

5) The perimeter of the triangle is 67 inches. What is the length of the third side?

Perimeter of a Triangle image 15 US

The perimeter (the total distance around the triangle) is the sum of the sides:

\begin{aligned} & 31 \, + \, 17 \, + \, ? = 67 \\\\ & 48 \, + \, ? = 67 \end{aligned}

To find the missing side, think about what number added to 48 is 67.

Since 48 + 19 = 67, the missing side length is 19 inches.

6) The side of a large garden, shaped like an equilateral triangle, is 23 meters. The side of a smaller garden, shaped like an equilateral triangle, is 18 meters. What is the difference in the perimeter of the two gardens?

Each side of an equilateral triangle is the same length, so all sides of the large garden are 23 meters and all sides of the smaller garden are 18 meters.

Large Garden: Small Garden:

Perimeter of a Triangle image 16 US

Perimeter: Perimeter:

23+23+23=69 meters \hspace{1.1cm} 18+18+18=54 meters

To find the difference between the perimeter of the large and small garden, subtract:

69 \, – \, 54 = 15

The difference in perimeter is 15 meters.

Perimeter of a triangle FAQs

Since all triangles have 3 sides, just add the length of all three sides to find the perimeter. There are additional ways to solve for different types of triangles, like equilateral and isosceles triangles. For equilateral triangles, multiply the length of one side by 3. For isosceles triangles, multiply one of the equal sides by 2 and then add the third side.

The area of a triangle can be calculated with the formula: \cfrac{1}{2} \times b \times h.

Yes, triangles can have measurements that include fractions or decimals. The perimeter is found the same way – by adding all the sides.

It is a theorem that relates to right-angled triangles and the relationship between the two sides that form a 90 -degree angle and the long side, called the hypotenuse. This is a topic covered in middle school.

The next lessons are

  • Prism shape

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Perimeter of a Triangle Calculator

What is the perimeter of a triangle, how to find the perimeter of the triangle the formula for a perimeter of a triangle, how to use our perimeter of a triangle calculator.

With our perimeter of a triangle calculator, you can easily calculate the perimeter of that figure. The tool has the basic formula implemented – the one assuming you know all three triangle sides. But that's not all - our calculator is better than the other ones you can find on the Internet because we've also implemented two other formulas for triangle perimeter, depending on the values you know. Isn't it awesome?

If you are still wondering how to find a triangle's perimeter or are curious about the formulas for a perimeter of a triangle behind this calculator, keep reading. Check out our other handy tools: triangle area calculator , right triangle calculator , and equilateral triangle calculator – these are a safe bet for your geometry problems.

The perimeter is a distance around the shape – in our case, around the triangle. You can think about it as a path surrounding this figure. In real-life problems, a perimeter of a triangle may be useful in making a fence around the triangular parcel, tying up a triangular box with ribbon, or estimating the lace needed for binding a triangular pennant. However, we guess that you will probably use it in your Maths class ;)

The basic formula is uncomplicated. Just add up the lengths of all the sides of the triangle, and you will obtain the perimeter value:

  • Formula given three sides (SSS)

However, you don't always have three sides given. What can you do then? In these cases, other equations derived from trigonometry may be used, depending on what you know about the triangle:

Two sides and the angle between them (SAS)

Use the law of cosines to find the third side and then the perimeter:

Two angles and a side between them (ASA)

Use the law of sines to find the remaining two sides and then the perimeter:

🙋 For an explanation of the law of sines and cosines, don't hesitate to visit our dedicated calculators: the law of cosines calculator and the law of sines calculator !

Let's take a practical application case as an example. Imagine that you want to make a small garden in your backyard and you want to calculate how much fence you need to enclose it.

  • Choose the formula, according to the data given . Assume that we know two sides of the triangle garden and the angle between them.
  • Enter the values into the proper boxes . Dimensions of our triangular garden are equal to a = 8  ft a= 8\ \text{ft} a = 8   ft , b = 6  ft b = 6\ \text{ft} b = 6   ft and the angle between them is γ = 75 ° \gamma = 75\degree γ = 75° . Remember that you can choose the unit by clicking on its name.
  • Tadaaaam! The perimeter of a triangle calculator displays the solution in a blink of an eye! In our case, we need 22.67  ft 22.67\ \text{ft} 22.67   ft of fence.

We are sure that after this detailed explanation and example, you understood what the perimeter of a triangle is. Keep practicing!

How do I find the perimeter of an SSS triangle?

To find the perimeter of a triangle knowing its three sides (SSS triangle), all you have to do is add the three known sides.

For example, the perimeter of a triangle with sides a = 3 cm , b = 2 cm and c = 4 cm can be calculated as follows:

perimeter_SSS = a + b+ c

perimeter_SSS = 3 cm + 2 cm + 4 cm

perimeter_SSS = 9 cm

What's the formula used to find the perimeter of SAS triangle?

To find the perimeter of a triangle knowing two sides ( a and b ) and the angle between them ( γ ) or a SAS triangle, we use the law of cosines to find the third side and then the perimeter. As a result, we can use the following expression and the perimeter formula for a SAS triangle:

perimeter_SAS = a + b + √(a 2 +b 2 - 2⋅a⋅b⋅cos(γ))

What's the formula used to find the perimeter of an ASA triangle?

To calculate the perimeter of a triangle knowing two angles ( β and γ ) and the side between them ( a ) or an ASA triangle, we use the law of sines to find the third side and then the perimeter. We express the perimeter formula for an ASA triangle as:

perimeter_ASA = a + a⋅(sin(β) + sin(γ)) / sin(β + γ)

Car crash force

Coffee kick, perimeter of a polygon.

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How to Find the Perimeter of a Triangle

Last Updated: August 28, 2023 Fact Checked

This article was co-authored by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. There are 11 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,787,133 times.

Finding the perimeter of a triangle means finding the distance around the triangle. [1] X Research source The simplest way to find the perimeter of a triangle is to add up the length of all of its sides, but if you don't know all of the side lengths you will need to calculate them first. This article will first teach you to find the perimeter of a triangle when you do know all three side lengths; this is the easiest and most common way. It will then teach you to find the perimeter of a right triangle when only two of the side lengths are known. Finally, it will teach you to find the perimeter of any triangle for which you know two side lengths and the angle measure between them (an "SAS Triangle"), using the Law of Cosines.

Finding the Perimeter When Three Side Lengths are Known

Step 1 Remember the formula for finding the perimeter of a triangle.

  • What this formula means in simpler terms is that to find the perimeter of a triangle, you just add together the lengths of each of its 3 sides.

Step 2 Look at your triangle and determine the lengths of the three sides.

  • This particular example is called an equilateral triangle, because all three sides are of equal length. But remember that the perimeter formula is the same for any kind of triangle.

Step 3 Add the three side lengths together to find the perimeter.

  • In another example, where a = 4 , b = 3 , and c=5 , the perimeter would be: P = 3 + 4 + 5 , or 12 .

Step 4 Remember to include the units in your final answer.

  • In this example, the side lengths are each 5cm, so the correct value for the perimeter is 15cm.

Finding the Perimeter of a Right Triangle When Two Sides are Known

Step 1 Remember what a right triangle is.

  • If, for example, you know that side a = 3 and side b = 4 , then plug those values into the formula as follows: 3 2 + 4 2 = c 2 .
  • If you know the length of side a = 6 , and the hypotenuse c = 10 , then you should set the equation up like so: 6 2 + b 2 = 10 2 .

Step 5 Solve the equation to find the missing side length.

  • In the first example, square the values in 3 2 + 4 2 = c 2 and find that 25= c 2 . Then calculate the square root of 25 to find that c = 5 .
  • In the second example, square the values in 6 2 + b 2 = 10 2 to find that 36 + b 2 = 100 . Subtract 36 from each side to find that b 2 = 64 , then take the square root of 64 to find that b = 8 .

Step 6 Add up the lengths of the three side lengths to find the perimeter.

  • In our first example, P = 3 + 4 + 5, or 12 .
  • In our second example, P = 6 + 8 + 10, or 24 .

Do you have the perimeter and are missing one side? Then you should subtract the sum of the two sides from the perimeter. This number equals the length of the missing side.

Finding the Perimeter of an SAS Triangle Using the Law of Cosines

Step 1 Learn the Law of Cosines.

  • For example, imagine a triangle with side lengths 10 and 12, and an angle between them of 97°. We will assign variables as follows: a = 10 , b = 12 , C = 97°.

Step 3 Plug your information into the equation and solve for side c.

  • c 2 = 10 2 + 12 2 - 2 × 10 × 12 × cos (97) .
  • c 2 = 100 + 144 – (240 × -0.12187) (Round the cosine to 5 decimal places.)
  • c 2 = 244 – (-29.25)
  • c 2 = 244 + 29.25 (Carry the minus symbol through when cos (C) is negative!)
  • c 2 = 273.25

Step 4 Use side length c to find the perimeter of the triangle.

  • In our example: 10 + 12 + 16.53 = 38.53 , the perimeter of our triangle!

Triangle Perimeter Calculator, Practice Problems, and Answers

problem solving perimeter of a triangle

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Calculate the Area of a Triangle

  • ↑ http://www.mathsisfun.com/geometry/perimeter.html
  • ↑ https://www.cuemath.com/measurement/perimeter-of-a-triangle/
  • ↑ https://www.omnicalculator.com/math/triangle-perimeter
  • ↑ https://www.varsitytutors.com/basic_geometry-help/how-to-find-the-perimeter-of-a-right-triangle
  • ↑ https://sciencing.com/perimeter-right-triangle-6196682.html
  • ↑ https://www.cuemath.com/measurement/perimeter-of-right-angled-triangle/
  • ↑ https://www.mathsisfun.com/algebra/trig-cosine-law.html
  • ↑ http://www.cimt.org.uk/projects/mepres/step-up/sect4/index.htm
  • ↑ https://www.calculatorsoup.com/calculators/geometry-plane/triangle-theorems.php
  • ↑ http://www.rapidtables.com/calc/math/Cos_Calculator.htm
  • ↑ https://www.mathsisfun.com/algebra/trig-solving-sas-triangles.html

About This Article

Grace Imson, MA

To find the perimeter of a triangle, use the formula perimeter = a + b + c, where a, b, and c are the lengths of the sides of the triangle. For example, if the length of each side of the triangle is 5, you would just add 5 + 5 + 5 and get 15. Therefore, the perimeter of the triangle is 15. If you only know the length of 2 of the triangle’s sides, you can still find the perimeter if it’s a right triangle, which means the triangle has one 90-degree angle. Just use the Pythagorean theorem, which is a^2+ b^2 = c^2, where a and b are the lengths of the known sides and c is the length of the unknown hypotenuse. For example, if the length of the known sides are 3 and 4, you would just add 3^2+ 4^2, or 9 + 16, and get 25. Then, you would take the square root of 25 to find c, which is 5. Therefore, the length of the unknown side is 5. Finally, add all of the side lengths together to find the perimeter. In this case you would add 3 + 4 + 5 and get 12. Therefore, the perimeter of the triangle is 12. If you want to learn how to solve the perimeter of your triangle if you only know 2 sides and an angle, keep reading the article! Did this summary help you? Yes No

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Unit 7: Area and perimeter

About this unit, count unit squares to find area.

  • Intro to area and unit squares (Opens a modal)
  • Measuring rectangles with different unit squares (Opens a modal)
  • Measuring area with partial unit squares (Opens a modal)
  • Creating rectangles with a given area 1 (Opens a modal)
  • Creating rectangles with a given area 2 (Opens a modal)
  • Find area by counting unit squares 7 questions Practice
  • Find area with partial unit squares 7 questions Practice
  • Create rectangles with a given area 4 questions Practice

Area of rectangles

  • Transitioning from unit squares to area formula (Opens a modal)
  • Finding missing side when given area (Opens a modal)
  • Counting unit squares to find area formula (Opens a modal)
  • Area of rectangles review (Opens a modal)
  • Transition from unit squares to area formula 7 questions Practice
  • Find a missing side length when given area 7 questions Practice
  • Perimeter: introduction (Opens a modal)
  • Perimeter of a shape (Opens a modal)
  • Find perimeter by counting unit squares (Opens a modal)
  • Finding perimeter when a side length is missing (Opens a modal)
  • Finding missing side length when given perimeter (Opens a modal)
  • Perimeter & area (Opens a modal)
  • Perimeter and unit conversion (Opens a modal)
  • Applying the metric system to perimeter (Opens a modal)
  • Perimeter review (Opens a modal)
  • Find perimeter by counting units 4 questions Practice
  • Find perimeter when given side lengths 7 questions Practice
  • Find a missing side length when given perimeter 4 questions Practice

Area of parallelograms

  • Area of a parallelogram (Opens a modal)
  • Area of parallelograms (Opens a modal)
  • Area of parallelograms 4 questions Practice
  • Find missing length when given area of a parallelogram 4 questions Practice

Area of triangles

  • Area of a triangle (Opens a modal)
  • Area of triangles (Opens a modal)
  • Area of triangle proof (Opens a modal)
  • Find base and height on a triangle 4 questions Practice
  • Area of right triangles 4 questions Practice
  • Area of triangles 7 questions Practice
  • Find missing length when given area of a triangle 4 questions Practice

Area of shapes on grids

  • Area of a triangle on a grid (Opens a modal)
  • Area of a quadrilateral on a grid (Opens a modal)
  • Areas of shapes on grids 4 questions Practice

Area of trapezoids & composite figures

  • Area of trapezoids (Opens a modal)
  • Area of kites (Opens a modal)
  • Finding area by rearranging parts (Opens a modal)
  • Area of composite shapes (Opens a modal)
  • Perimeter & area of composite shapes (Opens a modal)
  • Challenge problems: perimeter & area (Opens a modal)
  • Area of trapezoids 4 questions Practice
  • Area of composite shapes 4 questions Practice
  • Area challenge 4 questions Practice

Area and circumference of circles

  • Radius, diameter, circumference & π (Opens a modal)
  • Labeling parts of a circle (Opens a modal)
  • Radius, diameter, & circumference (Opens a modal)
  • Circumference review (Opens a modal)
  • Radius & diameter from circumference (Opens a modal)
  • Area of a circle (Opens a modal)
  • Area of circles review (Opens a modal)
  • Area of a circle intuition (Opens a modal)
  • Radius and diameter 7 questions Practice
  • Circumference of a circle 4 questions Practice
  • Area of a circle 7 questions Practice
  • Area of parts of circles 4 questions Practice

Advanced area with triangles

  • Area of equilateral triangle (Opens a modal)
  • Area of equilateral triangle (advanced) (Opens a modal)
  • Area of diagonal-generated triangles (Opens a modal)

SplashLearn

Perimeter of a Triangle – Definition, Formula, Examples, FAQs

What is the perimeter of a triangle, perimeter of a triangle formula, how to find the perimeter of a triangle, solved examples on perimeter of a triangle, practice problems on perimeter of a triangle, frequently asked questions on perimeter of a triangle.

The perimeter of a triangle is the sum of all its sides. In other words, the perimeter of a triangle is the total length of its boundary.

A triangle is a polygon , a closed, 2-dimensional shape with three sides, three angles, and three vertices. Its perimeter is calculated by adding the length of all the sides. A perimeter of a 2D shape is measured in linear units of measurement like inches , feet , yards, etc.

Related Games

Answer Questions Related to Triangles Game

Perimeter of a Triangle: Definition

The perimeter of a triangle can be defined as the sum of all sides of a triangle. 

A triangle has three sides. 

Perimeter of a Triangle $=$ Sum of three sides of the triangle

Related Worksheets

Rectangles and Triangles Worksheet

What is the formula for the perimeter of a triangle? Let’s find out.

Let a, b, and c be the lengths of the three sides of a triangle.

Perimeter of triangle $= a + b + c$

Perimeter of a triangle formula

Step 1: Note down the lengths of all three sides of the given triangle. Ensure that the lengths are in the same unit.

Step 2: Add the lengths of the three sides.

Step 3: The sum represents the perimeter of the given triangle. Assign the same unit to the perimeter as the length of the sides.

Perimeter of an Equilateral Triangle

A triangle with three equal sides and three congruent angles $(60^\circ)$ is known as an equilateral triangle .

Consider an equilateral triangle ABC whose each side measures “a” units.

Perimeter of an equilateral triangle $= a + a + a = 3a$

So, how do you find the perimeter of a triangle with three equal sides? Simply multiply the length of the side by 3!

Perimeter of Equilateral Triangle formula

Perimeter of an Isosceles Triangle

A triangle with two equal sides is known as an isosceles triangle .

Since two sides of a triangle are equal, we have $a = b$

Perimeter of an isosceles triangle formula

Perimeter of an isosceles triangle $= a + a + c = 2a + c$

Perimeter of a Scalene Triangle

A triangle in which all the sides have different lengths is known as a scalene triangle . 

If the lengths of three sides are given by a, b, and c, then 

Perimeter $= a + b + c$

Perimeter of a scalene triangle formula

Perimeter of a Right-angled Triangle

A triangle in which one interior angle is $90^\circ$ is called a right triangle. The side opposite to the $90^\circ$ angle is called a hypotenuse. The other two sides are termed as “legs” of the right triangle. A right triangle has a base (b), hypotenuse (h), and perpendicular (p) as its sides. 

Perimeter of a right triangle

Going by Pythagoras’ theorem, we know that

$h^2 = p^2 + b^2$

Therefore, the perimeter of a right angle triangle $= b + p + h$

$= b + p + \sqrt{p^2 + b^2}$

Perimeter of an Isosceles Right Triangle

A right triangle in which the base and the height (two legs) are of equal length is called an isosceles right triangle . 

Perimeter of Isosceles Right Triangle formula

Applications of Perimeter of a Triangle

Take a look at the triangular park. Supposing we have to fence it along the outer boundary, we need to find the length of the fence required to cover the park. How do we do that? The length of the fence required can be calculated by finding the total length of the boundary (perimeter) of the triangle.

Perimeter of a triangular park

1. Find the perimeter of an equilateral triangle with side 6 inches. 

Solution: 

Side of an equilateral triangle $= a = 6$ inches

Perimeter of an equilateral triangle $= 3a$

$=3 \times 6$

$= 18$ inches

2. What is the perimeter of an isosceles triangle whose equal sides are 4 feet each and the unequal side is 6 feet?

Let equal sides be $a = 4$ feet and non equal side be $b = 6$ feet

Perimeter of the given isosceles triangle $= 2a+b$

                                                      $=(2 \times 4)+ 6$

$= 14$ feet

3. If the perimeter of the given triangle is 34 feet, what is the length of the missing side?

triangle XYZ with sides 9 ft, 12 ft and a missing length

In the $\Delta XYZ$, we have

$XY = 9$ feet

$XZ = 12$ feet

Perimeter $= 34$ feet …given

Perimeter of the triangle $= XY + YZ + XZ$

$34 = 9 +  12 + YZ$

$34 = YZ + 21$

$YZ = 13$ feet

4. What is the value of x if the perimeter of the triangle PQR is 40 units? Also, find the length of three sides.

A triangle with sides x, x+4, x+6

Solution:  

Perimeter of triangle $PQR = PQ + QR + PR$

$40 = x + x + 4 + x + 6$

$40 = 3x + 10$

Let’s find the length of the sides.

$PQ = x = 10$ units

$QR = x + 6 = 10 + 6 = 16$ units

$PR = x + 4 = 10 + 4 = 14$ units

5. If the base and height of a right isosceles triangle is 6 inches each, what would be its perimeter? 

Solution : $b = p = 6$ inches

$h = \sqrt{p^2 + b^2} = \sqrt{6^2 + 6^2} = \sqrt{36+36} = \sqrt{72} = 6\sqrt{2}$ inches

Perimeter of a right triangle$ = b + p + h = 6 + 6 + 6\sqrt{2} = (12 + 6\sqrt{2})$ inches

Perimeter of a Triangle – Definition, Formula, Examples, FAQs

Attend this quiz & Test your knowledge.

What is the perimeter of $\Delta ABC$?

Perimeter of a Triangle – Definition, Formula, Examples, FAQs

The perimeter of an equilateral triangle is 45 inches. What is the side of the triangle?

If the perimeter of a scalene triangle is 32 feet. if the two sides are 12 feet and 7 feet, then what is the third side, find the perimeter of a right angled isosceles triangle whose two sides are 4 inches each..

What is the difference between area and perimeter?

Area is the region bounded within the sides of a 2D shape, whereas perimeter is the total length of the boundary of the shape.

What is the formula for the area of a triangle?

Area of a Triangle $= \frac{1}{2} \times base \times height$

What is the semiperimeter of a triangle?

Semiperimeter is the half of the perimeter of a triangle. If the sides of a triangle are a,b and c, then the semiperimeter $= \frac{a + b + c}{2}$

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Perimeter of a Triangle

Learn about the perimeter of a triangle., perimeter of a triangle lesson, the perimeter formula.

A triangle is a polygon with three sides and three vertices. The formula for perimeter is given as:

  • P = a + b + c

Where P is the perimeter, and a , b , and c are the three side lengths.

perimeter-of-a-triangle-equation-and-diagram

Perimeter of a Triangle Example Problems

Let's go through a couple of example problems to practice finding the perimeter of a triangle.

Example Problem 1

What is the perimeter of a triangle with side lengths of 5, 6, and 3?

  • Let’s plug the side lengths into the perimeter formula.
  • P = 5 + 6 + 3 = 14
  • The perimeter of the triangle is 14.

Example Problem 2

An equilateral triangle has a perimeter measured to be 81 centimeters. What are the side lengths?

  • Since an equilateral triangle has three equal sides, a = b = c. We can substitute a + b + c with 3a.
  • Plugging what we know into the perimeter formula, we get: P = a + b + c 81 = 3a a = 27, b = 27, c = 27
  • The three sides each have a length of 27 centimeters.

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Pre-Algebra : Perimeter of a Triangle

Study concepts, example questions & explanations for pre-algebra, all pre-algebra resources, example questions, example question #1 : perimeter of a triangle.

problem solving perimeter of a triangle

Example Question #2 : Perimeter Of A Triangle

problem solving perimeter of a triangle

Not enough information is given.

problem solving perimeter of a triangle

Example Question #3 : Perimeter Of A Triangle

A right triangle has two short sides with lengths 5 and 12. What is the perimeter of the triangle?

To begin, first find the length of the third side of the triangle. We are given that the two short sides of the triangle have lengths of 5 and 12. Use the Pythagorean Theorem to find the length of the third side:

problem solving perimeter of a triangle

Now that we have the lengths of all three sides, add them together to find the perimeter:

problem solving perimeter of a triangle

The perimeter of the triangle is 30.

Example Question #4 : Perimeter Of A Triangle

Solve for the perimeter of the triangle.

Tom's, Bob's, and Fred's houses are in a triangle.

problem solving perimeter of a triangle

What is the perimeter of the triangle between their houses?

problem solving perimeter of a triangle

Example Question #1 : Properties Of Triangles

problem solving perimeter of a triangle

It cannot be determined from the information given.

problem solving perimeter of a triangle

Using the Pythagorean Theorem, the length of the second leg can be determined.

problem solving perimeter of a triangle

We are given the length of the hypotenuse and one leg.

problem solving perimeter of a triangle

The perimeter of the triangle is the sum of the lengths of the sides.

problem solving perimeter of a triangle

Find the perimeter of the triangle above.

Note: Figure not drawn to scale.

problem solving perimeter of a triangle

None of these answers are correct.

problem solving perimeter of a triangle

Because the lengths are in inches, the answer must be in inches as well.

Example Question #6 : Perimeter Of A Triangle

problem solving perimeter of a triangle

Not enough information given to solve.

problem solving perimeter of a triangle

Example Question #7 : Perimeter Of A Triangle

problem solving perimeter of a triangle

For an isosceles triangle, if two of the sides are 3 and 6, which of the following is a possible perimeter?

problem solving perimeter of a triangle

The perimeter is the sum of the three sides.

problem solving perimeter of a triangle

Example Question #9 : Perimeter Of A Triangle

What is the perimeter of an equilateral triangle with a length of 5?

problem solving perimeter of a triangle

There are three equal sides in an equilateral triangle.  

problem solving perimeter of a triangle

Substitute the side length.

problem solving perimeter of a triangle

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problem solving perimeter of a triangle

Home / United States / Math Classes / 3rd Grade Math / Perimeter of a Triangle

Perimeter of a Triangle

A triangle is a three-sided polygon. The perimeter is the sum of the lengths of the boundary of a figure. So, the perime ter of a triangle is calculated by adding all three sides together. We will learn about the perimeter of a triangle and solve some examples for a better understanding of this concept. ...Read More Read Less

Table Of Contents

problem solving perimeter of a triangle

What is a polygon?

What is a triangle, what is the perimeter, the perimeter of a triangle, rapid recall, solved examples.

  • Frequently Asked Questions

Figures can be open or closed, which may be made up of straight lines and curved lines, or made up of only curves, or only lines. If a figure is closed and made up of only line segments, it is called a polygon .

1

A triangle is a closed polygon made up of three line segments. So, a triangle has three sides, three vertices, and three angles. In geometry, a triangle is denoted by “△”. If there is a triangle whose vertices are A, B, and C, we can denote the triangle as △ABC.

2

The perimeter of any figure is the total length of its boundary. And the perimeter of a polygon is calculated by adding all the sides of the polygon together. Since the perimeter is actually the length of the boundary of a closed shape, the unit of the perimeter is the same as that of the units of length, such as centimeter(cm), millimeter(mm), inch(in), feet(ft), yard(yd), kilometer(km), and mile(mi).

The perimeter of a triangle is defined as the total length of the boundary. Since the boundary of the triangle is made by three line segments which are its sides , the perimeter of the triangle is equal to the sum of all three sides. If there is a triangle whose side lengths are a, b, and c, the perimeter of the triangle is equal to, a + b + c.

3

The perimeter of a △ABC =  sum of all three sides

Hence, perimeter P = a + b + c

The perimeter of different types of triangles:

4

Example 1: Find the perimeter of an equilateral triangle whose side length is 7 inches.

The perimeter of an equilateral triangle, P = 3a

P = 3 \(\times\) 7

So, the perimeter of the equilateral triangle is 21 inches.

Example 2: Find the perimeter of an isosceles triangle whose equal side lengths are 5 feet and the unequal side length is 7 feet.

The perimeter of an isosceles triangle, P = 2a + b

P = 2 \(\times\) 5 + 7

So, the perimeter of the isosceles triangle is 17 feet.

Example 3: Find the perimeter of a triangle whose sides are 3 cm, 4 cm, and 5cm.

The perimeter of a triangle P = a + b + c

P = 3 + 4 + 5

So, the perimeter of the triangle is 12 centimeters.

How many vertices are there in a triangle?

A triangle has three vertices.

How many equal sides are there in an isosceles triangle?

There are two equal sides in an isosceles triangle.

How can we find the perimeter of a triangle?

A triangle is made up of three line segments, so the sum of all three sides of a triangle gives us its perimeter.

Name the triangle whose interior angle is 60°.

An equilateral triangle has each angle equal to 60°.

What is the formula for the perimeter of the isosceles triangle?

P = 2a + b , where P is the perimeter of the isosceles triangle, a is the length of the equal sides, and b is the length of the third side.

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Perimeter of a Triangle Worksheets

Navigate through our meticulously designed perimeter of a triangle worksheet pdfs for students of grade 2 through grade 8, to develop skills like finding the perimeter of a triangle with dimensions represented as integers, decimals and fractions. Determine the measure of the sides using the perimeter and solve word problems too. Click on our free worksheets and kick-start practice!

Integers | Type 1 - Level 1

Perimeter of a Triangle | Integers - Type 1

Utilize this assortment of worksheets focusing on finding the perimeter of a triangle whose dimensions are given as integers ≤ 20 in level 1 and ≥ 10 in Level 2, to provide sufficient practice to 2nd grade, 3rd grade, and 4th grade children.

  • Download the set

Integers | Type 2 - Level 1

Perimeter of a Triangle | Integers - Type 2

Bolster practice in finding the perimeter of equilateral, isosceles and scalene triangles presented as geometric shapes and in word format. Add up the side lengths to compute the perimeter.

Perimeter of a Triangle | Decimals – Type 1

Perimeter of a Triangle | Decimals – Type 1

This compilation of 5th grade printable perimeter of a triangle worksheets features 40+ triangles as geometric figures with decimal dimensions. Work out the perimeter by adding up the side lengths.

Perimeter of a Triangle | Decimals – Type 2

Perimeter of a Triangle | Decimals – Type 2

Supplement your practice finding the perimeter of a triangle with decimal side lengths with this set of two-fold exercises offering the dimensions on figures as well as in word format.

Perimeter of a Triangle | Fractions – Type 1

Perimeter of a Triangle | Fractions – Type 1

Calculate the perimeter by plugging in the side lengths given as fractions or mixed numbers in the formula P = a + b + c, where a, b, c are the three sides of the triangle. Convert to mixed numbers if required.

Perimeter of a Triangle | Fractions – Type 2

Perimeter of a Triangle | Fractions – Type 2

Drum into the heads grade 4 and grade 5 children the formula for finding the perimeter of triangles with pdf worksheets! All they need to do is add up the fractional side lengths and simplify the sum.

Perimeter of a Triangle | Congruent Property

Perimeter of a Triangle | Congruent Property

Figure out the perimeter of equilateral and isosceles triangles in this batch of printable worksheets for grade 6 and grade 7. The measures and the congruent sides are indicated in each triangle, apply the congruence property to solve.

Find the Missing Side of a Triangle using the Perimeter

Find the Missing Side of a Triangle using the Perimeter

Included here are scalene, equilateral and isosceles triangles with their congruent parts marked. Rearrange the perimeter formula, substitute the known values and solve for the missing side of each triangle.

Find the Sides of a Triangle | Algebra in Triangles

Find the Sides of a Triangle | Algebra in Triangles

The dimensions are presented as algebraic expressions to provide ample practice for 6th grade, 7th grade, and 8th grade students. Add up the side lengths and equate with the given perimeter. Solve for 'x', plug its value in the expression to find the length of each side of the triangle.

Perimeter of a Triangle Word Problems

Perimeter of a Triangle Word Problems

Read each real-life scenario carefully, highlight the measures, visualize the triangle and then determine its perimeter. Reiterate the concept of finding the perimeter of triangles with this set of printable word problems worksheets.

Related Worksheets

» Perimeter of Squares

» Perimeter of Rectangles

» Perimeter of Quadrilaterals

» Perimeter of Polygons

» Triangles

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Perimeter of a triangle

The perimeter of a triangle is the distance around it, which is the sum of the lengths of its sides. The formula for the perimeter of a triangle is:

Perimeter = a + b + c

where a, b, and c are the lengths of the three sides of the triangle.

problem solving perimeter of a triangle

Special triangles

Isosceles triangles , equilateral triangles , and right triangles have a number of relationships that allow us to find their perimeters without necessarily knowing all of their side lengths. Below are the formulas for the perimeters of these triangle types.

Isosceles triangle

Perimeter = 2 × l + b

Where l is the side length and b is the base length.

problem solving perimeter of a triangle

Equilateral triangle

Perimeter = 3 × s

Where s is the side length

problem solving perimeter of a triangle

Right triangle

You can use the Pythagorean Theorem to find the perimeter of a right triangle if you know, or can determine, the lengths of at least two sides from the given information.

problem solving perimeter of a triangle

Referencing the triangle above:

If a and b are given,

If a and c are given,

If b and c are given,

What is the perimeter of a triangle with a hypotenuse c = 26, and a side length a = 10?

Perimeter of a Triangle

We will discuss here how to find the perimeter of a triangle. We know perimeter of a triangle is the total length (distance) of the boundary of a triangle.

Perimeter of a triangle is the sum of lengths of its three sides.

Perimeter of a Triangle

For example, perimeter of the ∆PQR = PQ + QR + RP

● The perimeter of a triangle ABC

Perimeter of a Triangle

                    = AB + BC + CA

                    = 2 cm + 4 cm + 3 cm,

(add the length of each side of the triangle).

                     = 9 cm

Perimeter of the triangle = Sum of the sides.

Perimeter of a Triangle

●  A triangle has 3 sides

The perimeter of the triangle XYZ

= 3 cm + 5 cm + 4 cm

The perimeter of the triangle = Sum of the lengths of three sides.

Let us consider some of the examples on perimeter of a triangle:

1. Find the perimeter of a triangle having sides 3 cm, 8 cm and 6 cm.

Examples on Perimeter of a Triangle

Perimeter of a triangle

                   = Sum of all the three sides

                   = AB + BC + AC

                   = 3 cm + 8 cm + 6 cm

                   = 17 cm

2. Find the perimeter of the triangle PQR whose sides are 4 cm, 6 cm and 8 cm.

Perimeter of Triangle

In the figure PQ = 4 cm, PR = 6 cm and QR = 8 cm

The perimeter of the rectangle PQR

                            = 4 cm + 6 cm + 8 cm

                            = 18 cm

3. Find the perimeter of an equilateral triangle whose one side is 5 cm.

A triangle in which all the sides are equal is called an equilateral triangle.

Perimeter of the equilateral triangle = 3 × side

                                                    = 3 × 5 cm

                                                    = 15 cm

Thus, perimeter = 15 cm.

4.  Find the perimeter of a triangle whose length of three sides are 8 cm, 11 cm, 13 cm.

Solution: To find the perimeter of the triangle, we add all the sides together.

                  = Sum of all the three sides

                  = 8 cm + 11 cm + 13 cm

                  = 32 cm

5. Find the perimeter of a triangle whose sides are 5 cm, 2 cm and 3 cm.

Perimeter of the triangle is the sum of the lengths of its sides.

Perimeter = 5 cm + 2 cm + 3 cm

Thus, perimeter = 10 cm.

6. Find the perimeter of each triangle.

Find the Perimeter of each Triangle

Solution: 

(i) Perimeter of ∆XYZ = 5.5 cm + 6 cm + 6 cm = 17.5 cm

(ii) Perimeter of ∆ABC = 8 cm + 6 cm + 6 cm = 20 cm

(iii) Perimeter of ∆PQR = 4 cm + 3 cm + 5 cm = 12 cm

7. Find the perimeter of the given shapes.

Find the Perimeter of the given Shapes

(i) Perimeter = PQ + QR + RS + ST + TU + UV + VP

                   = 2.5 cm + 3 cm + 2 cm + 3 cm + 2.5 cm + 4 cm + 4 cm

                   = 21 cm

(ii) Perimeter = PQ + QR + RS + SP

                    = 4 cm + 4 cm + 4 cm + 4 cm

                    = 16 cm

(iii) Perimeter = PQ + QR + RS + ST + TP

                     = 7 cm + 6 cm + 4 cm + 3 cm + 5 cm

                     = 25 cm

Word Problems on Perimeter of a Triangle:

1. Two sides of a triangle are 3 cm and 4 cm. Find the third side of the triangle if its perimeter is 11 cm.

First side of the triangle                            = 3 cm

Second side of the triangle                        = 4 cm

Perimeter of the triangle                           = Sum of the lengths of sides

i.e. sum of the lengths of the sides            = 11 cm

3 cm + 4 cm + length of the third side      = 11 cm

7 cm + length of the third side                 = 11 cm

But we know that 7 cm + 4 cm                = 11 cm ( Note: 11 – 7 = 4 )

Therefore, length of the third side            = 4 cm

Questions and Answers on Perimeter of a Triangle:

1. A triangle has a perimeter of 50 cm. If its two sides are of lengths 15 cm and 19 cm, what is the length of the third side?

Answer:  16 cm

●  Related Concepts

● Units for Measuring Length

● Measuring Instruments

● To Measure the Length of a Line-segment

● Perimeter of a Figure

● Perimeter of a Triangle

● Perimeter of a Rectangle

● Perimeter of a Square

● Unit of Mass or Weight

● Examples on Unit of Mass or Weight

● Units for The Measurement of Capacity

● Examples on Measurement of Capacity

● Measurement of Time

● Read a Watch or a Clock

● Antemeridian (a.m.) or Postmeridian (p.m.)

● What Time it is?

● Time in Hours and Minutes

● 24 Hour Clock

● Units of Time

● Examples Units of Time

● Time Duration

● Reading and Interpreting a Calendar

● Calendar Guides us to Know

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Area and Perimeter of a Triangle

Introduction.

In this article, we learn all about triangles. Starting with the definition and properties of the shape, we learn through intuitive backgrounds before we finally measure the dimensions of the shape through its sides and how much area a triangle takes. While doing so, we will also try some examples to guide us in learning. 

What is a Triangle?

A triangle , by definition, is a shape that has three sides . When we think of a triangle, we can associate it with road signs, pyramids, flags, and even musical instruments!

In geometric terms, a triangle consists of a set of three straight lines or sides , three interior angles that are formed by the sides, and three vertices that are intersections of the lines forming the triangle.

problem solving perimeter of a triangle

What are the Types of Triangles?

We mostly recognize triangles as having three sides. But what are the specific types that a triangle can take, and how can we differentiate them? Apparently, triangles can be described based on either the characteristics of their sides or their interior angles.

Based on Side Length

Triangles can be classified according to the length of their sides. They can either be an equilateral triangle, an isosceles triangle , or a scalene triangle .

An equilateral triangle is a triangle wherein all sides have equal length .

On the other hand, an isosceles triangle is a triangle with two sides of equal length .

If a triangle has all sides with different lengths , it is said to be a scalene triangle .

Based on Interior Angles

Triangles can also be classified based on the measure of their interior angles. There are three types according to the interior angles of a triangle: acute triangles , oblique triangles, and right triangles.

An acute triangle is a triangle whose two interior angles measure less than 90°.

On the other hand, an oblique triangle is a triangle with one interior angle measuring less than 90°.

Then, a right triangle is a triangle with one interior angle measuring exactly 90°. 

What is a Perimeter?

We define the perimeter of a two-dimensional shape as a measure of the boundary enclosing it .

problem solving perimeter of a triangle

In an intuitive approach, we can relate the idea of a perimeter to how far we jog through a lap in the park. If we begin measuring the distance from the start of the lap, and then trace a closed path until we get back to the starting point, we can measure the perimeter of the park through the lap we have run across:

Thus, for any closed shape we can get its perimeter by taking the length of the edges enclosing the shape.

What is an Area?

On the other hand, we can define the area of a two-dimensional shape as a measure of the space it occupies in the two-dimensional plane .

Intuitively, we can think of shapes in a plane as objects in a room. Each object takes up some space in the room, depending on the kind of object and its size. In this sense, we can measure a shape’s area through the shape’s dimensions.

Perimeter of a Triangle

From the definitions provided earlier, we can now discuss in detail the formula for solving the perimeter of a triangle. We first derive the basic formula involved, provide intuitive insights, and then show how we apply this formula with some examples.

Derivation of the Perimeter Formula of a Triangle

We know that the perimeter can be defined as the sum of the lengths of all edges covering a shape. In a triangle, we have three edges. Hence, we can say that the perimeter of a triangle can be expressed as the sum of the length of each side of the triangle:

problem solving perimeter of a triangle

Perimeter=Length of 1 st Side+Length of 2 nd Side+Length of 3 rd Side

If we denote each side as a, b, and c, we can compress this formula into a shorter form:

Perimeter=a+b+c

What is the Perimeter Formula of a Triangle?

From the previous derivation performed, the perimeter P of a triangle whose sides a, b, c are known is given by:

We note that the unit of the perimeter is expressed in terms of the same units as with the sides given.

How is it Used?

As a practice on how to apply the formula, let us work together on finding the perimeter of the triangle shown below:

problem solving perimeter of a triangle

From the given figure, we set the order of the sides arbitrarily and assign the given values to each side. Hence, we can say that:

a=4 units b=3 units c=5 units

Moreover, since the lengths of all sides are known, we can use the Perimeter Formula for a triangle:

We then substitute the given lengths to the formula to get:

P=4 units+3 units+5 units

Afterward, we add the numbers to obtain the value of the perimeter:

We then conclude that the perimeter of the triangle is 12 units. This quantity is expressed in terms of the same units as the radius.

Area of a Triangle

After learning about the Perimeter Formula of a triangle, we can furthermore measure the area of a triangle based on the dimensions of the said shape.

Derivation of the Area Formula of a Triangle

We first consider one side of a triangle, preferably its longest side. We call this side the base of the triangle:

problem solving perimeter of a triangle

Then, we form a line that is perpendicular to this side and passes through one of the vertices of the triangle. This is called the altitude of the triangle:

problem solving perimeter of a triangle

If we reimagine the triangle as a right triangle whose sides are given by the base and the altitude, we have the following figure:

problem solving perimeter of a triangle

Now, suppose we superimpose a rectangle whose length is equal to the base of the triangle, and whose height is equal to the altitude of the triangle:

problem solving perimeter of a triangle

From the Area Formula of a rectangle , we know that the area of this rectangle is given by:

A rectangle =base×altitude

Upon observation, we see that the reimagined triangle has an area that is half the area of the superimposed rectangle. Hence, we can say that:

Area=½ x A rectangle

If we substitute the Area Formula for the rectangle, we then arrive at the Area Formula for a triangle:

Area=½ x base x altitude

What is the Area Formula of a Triangle?

Through the derivation we have previously done, the area A of a triangle whose base b and altitude h is known can be expressed using the formula:

We note that for both formulas, the unit of the area is expressed in terms of squared units of the given radius/diameter.

Again, we practice what we have learned so far by working on a guided example for finding the area of the triangle:

problem solving perimeter of a triangle

From the given figure, we are given the base of the triangle to be 6 units, and its altitude to be 2 units. Hence, we can say that:

b=6 units h=2 units

Moreover, since the base and altitude are known, we can use the Area Formula of a triangle:

We then substitute the given base and altitude to the formula, and we get:

A=½ x 6 units × 2 units

Afterward, we multiply the numbers together to obtain the area of the triangle :

A=6 units 2

We then conclude that the area of the triangle is 6 units 2 . This quantity is expressed in terms of squared units of the radius.

Problem-Solving Examples

We can now proceed to solve sample problems to apply what we have learned so far. Each problem tackles different formulas discussed and gives us a challenge on how to solve through the information given to us.

Sample Problem 1:

What is the perimeter of an isosceles triangular glass pane whose sides are given 4 cm 4 cm, and 3 cm?

We recall that the perimeter P of a triangle is given by the formula:

With the given side lengths a=b=4 cm and c=3 cm, we substitute these values into the formula:

P=4 cm+4 cm+3 cm

Finally, by adding the lengths we obtain the value of the perimeter to be:

Therefore, we conclude that the perimeter of the triangular glass pane is 11 cm.

Sample Problem 2:

An athlete is training for her triathlon. In a triangular lap, she runs for twelve kilometers, swims for five kilometers, then races through thirteen kilometers by bike. How much distance does she train through a single lap?

We can first assign the sides of the triangular lap with each part of the triathlon. For this problem, we use the following values:

a=12 km b=5 km c=13 km

Then, the perimeter P of a triangle is given by the formula:

Hence, we can substitute the given values to get:

P=12 km+5 km+13 km

Lastly, we take the sum of the sides to determine the perimeter of the lap:

Therefore, we conclude that the athlete trains for is 30 kilometers.

Sample Problem 3:

Suppose we have a triangle whose third side is unknown. The lengths of the two sides are given by a=10 in and b=15 in. We are then asked the following:

  • What is the length of the third side?
  • What type of triangle is formed by the three sides?
  • What is the perimeter of the triangle?
  • For this problem, we are given two sides and an unknown third side. By the Pythagorean Theorem, we can solve for the length of the unknown side c using the given sides a and b:

c=$\sqrt{a^2+b^2}$

Substituting the given lengths, we have:

c=$\sqrt{{10 in}^2+ (15 in)^2}$

Taking the square of both terms inside the radical sign, we expand the sum we wish to solve:

c=$\sqrt{100 in^2+225 in^2}$

Adding the two numbers inside the radical, we simplify the expression into:

c=$\sqrt{325 in^2}$

Finally, we take the square root of the number inside the radical to get the length of the unknown side, then round it off to the nearest whole number:

Therefore, the length of the third side is 18 inches.

  • Since each side has a different length, we conclude that the triangle formed by these sides is a scalene triangle .
  • We recall that the Perimeter Formula for a triangle is given by:

Using the given side lengths a=10 in and b=15 in, along with the computed length of the third side c=18 in, we substitute these values into the equation to get:

P=10 in+15 in+18 in

Adding together these numbers, we then obtain the value of the perimeter:

Therefore, we conclude that the perimeter of the triangle is 43 inches.

Sample Problem 4:

What is the area of a right triangle whose base is 10 units and has an altitude of 18 units?

We recall that the area A of a triangle is given by the formula:

With the given base b=10 u, and an altitude of h=18 u, we substitute these into the formula to get:

A=½ x 10 units x 18 units

Finally, by multiplying the three numbers we can compute the value of the area:

A=90 units 2

Hence, the area of the right triangle is 90 squared units.

Sample Problem 5:

In the figure below, we are given an isosceles triangle whose sides are shown:

problem solving perimeter of a triangle

  • What is the length of the base of the triangle?
  • What is the length of the altitude of the triangle?
  • What is the area of the triangle?
  • We first observe that in the given figure, the base of the triangle can be expressed as the sum of the lengths of x and y:

Substituting the given side lengths x=y=7 cm, we get the value of b:

b=7 cm+7 cm=14 cm

Therefore, the length of the base is 14 cm.

  • We note that the altitude forms two right triangles. To solve for the length of the altitude, we apply the Pythagorean Theorem , using the sides a, x, and the altitude h:

a 2 =x 2 +h 2

Re-writing the equation in terms of the altitude, we have:

h=$\sqrt{a^2-x^2}$

Then, we proceed by substituting the known lengths:

h=$\sqrt{(25 cm)^2-(7 cm)^2}$

Taking the squares of both numbers inside the radical, we get:

h=$\sqrt{625 cm^2-49 cm^2}$

Afterward, we take the difference between the two squares:

h=$\sqrt{576 cm^2}$

Thus, the value of the unknown h can be obtained by taking the square root of the difference obtained earlier:

Therefore, the length of the altitude of the triangle is 24 cm.

  • We recall that the Area Formula for a triangle is given by:

Using the values obtained from Part A and Part B, we know the base and the altitude of the triangle to be:

b=14 cm h=24 cm

Substituting these values into the formula, we get:

A=½×14 cm×24 cm

Finally, multiplying the numbers together we get the value of the area:

Therefore, we conclude that the area of the isosceles triangle is 168 cm 2 .

Sample Problem 6:

Suppose a painter wants to paint a solid red triangle covering half of a 15 ft×15 ft square wall. For every liter of paint, an area of 100 square feet can be painted. How much red paint does the painter need to finish painting the triangle?

We first note that the painter is tasked to paint a right isosceles triangle with red paint. Therefore, we know that the base and the altitude of the triangle are both equal to the length of the side of the square wall:

Then, we solve for the area of the triangle to be painted. Using the Area Formula of a triangle, we have:

Substituting the values we have obtained earlier, we get the product of three numbers:

A=½×15 ft×15 ft

As such, the area of the triangle can be obtained:

A=112.5 ft 2

To determine how much red paint the painter needs, we apply the given ratio of 1 liter :100 ft 2 , as shown below:

1 l=100 ft 2

For an area A=112.5 ft 2 , the amount of red paint R required is given by:

R=112.5 ft 2 x ($\frac{1l}{100 ft^2}$)=1.125 liters

Therefore, the painter needs 1.125 liters of red paint to finish painting the red triangle.

A triangle is a shape that has three sides . It is also formed by a set of three interior angles , formed between the sides of the triangle, and three vertices , or intersections of the sides of the triangle.

A triangle can be classified according to the following characteristics:

Based on side length : an equilateral triangle has all sides equal, an isosceles triangle has two equal sides, and a scalene triangle has no equal sides,

Based on interior angle measure : an acute triangle has two interior angles measuring less than 90°, an oblique triangle has one interior angle measuring less than 90°, and a right triangle has one angle measuring exactly 90°.

A perimeter is a measure of the boundary enclosing a shape . It can be obtained by taking the length of the edges enclosing the shape.

On the other hand, an area is a measure of the space a shape occupies in a 2D plane . This quantity is dependent on the dimensions of the shape.

The perimeter P of a triangle with known sides a, b, c is given by the following formula:

The area A of a triangle whose base b and altitude h is known can be determined using the formula:

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  • 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
  • Introduction
  • 1.1 Introduction to Whole Numbers
  • 1.2 Use the Language of Algebra
  • 1.3 Add and Subtract Integers
  • 1.4 Multiply and Divide Integers
  • 1.5 Visualize Fractions
  • 1.6 Add and Subtract Fractions
  • 1.7 Decimals
  • 1.8 The Real Numbers
  • 1.9 Properties of Real Numbers
  • 1.10 Systems of Measurement
  • Key Concepts
  • Review Exercises
  • Practice Test
  • 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
  • 2.2 Solve Equations using the Division and Multiplication Properties of Equality
  • 2.3 Solve Equations with Variables and Constants on Both Sides
  • 2.4 Use a General Strategy to Solve Linear Equations
  • 2.5 Solve Equations with Fractions or Decimals
  • 2.6 Solve a Formula for a Specific Variable
  • 2.7 Solve Linear Inequalities
  • 3.1 Use a Problem-Solving Strategy
  • 3.2 Solve Percent Applications
  • 3.3 Solve Mixture Applications
  • 3.5 Solve Uniform Motion Applications
  • 3.6 Solve Applications with Linear Inequalities
  • 4.1 Use the Rectangular Coordinate System
  • 4.2 Graph Linear Equations in Two Variables
  • 4.3 Graph with Intercepts
  • 4.4 Understand Slope of a Line
  • 4.5 Use the Slope-Intercept Form of an Equation of a Line
  • 4.6 Find the Equation of a Line
  • 4.7 Graphs of Linear Inequalities
  • 5.1 Solve Systems of Equations by Graphing
  • 5.2 Solving Systems of Equations by Substitution
  • 5.3 Solve Systems of Equations by Elimination
  • 5.4 Solve Applications with Systems of Equations
  • 5.5 Solve Mixture Applications with Systems of Equations
  • 5.6 Graphing Systems of Linear Inequalities
  • 6.1 Add and Subtract Polynomials
  • 6.2 Use Multiplication Properties of Exponents
  • 6.3 Multiply Polynomials
  • 6.4 Special Products
  • 6.5 Divide Monomials
  • 6.6 Divide Polynomials
  • 6.7 Integer Exponents and Scientific Notation
  • 7.1 Greatest Common Factor and Factor by Grouping
  • 7.2 Factor Trinomials of the Form x2+bx+c
  • 7.3 Factor Trinomials of the Form ax2+bx+c
  • 7.4 Factor Special Products
  • 7.5 General Strategy for Factoring Polynomials
  • 7.6 Quadratic Equations
  • 8.1 Simplify Rational Expressions
  • 8.2 Multiply and Divide Rational Expressions
  • 8.3 Add and Subtract Rational Expressions with a Common Denominator
  • 8.4 Add and Subtract Rational Expressions with Unlike Denominators
  • 8.5 Simplify Complex Rational Expressions
  • 8.6 Solve Rational Equations
  • 8.7 Solve Proportion and Similar Figure Applications
  • 8.8 Solve Uniform Motion and Work Applications
  • 8.9 Use Direct and Inverse Variation
  • 9.1 Simplify and Use Square Roots
  • 9.2 Simplify Square Roots
  • 9.3 Add and Subtract Square Roots
  • 9.4 Multiply Square Roots
  • 9.5 Divide Square Roots
  • 9.6 Solve Equations with Square Roots
  • 9.7 Higher Roots
  • 9.8 Rational Exponents
  • 10.1 Solve Quadratic Equations Using the Square Root Property
  • 10.2 Solve Quadratic Equations by Completing the Square
  • 10.3 Solve Quadratic Equations Using the Quadratic Formula
  • 10.4 Solve Applications Modeled by Quadratic Equations
  • 10.5 Graphing Quadratic Equations in Two Variables

Learning Objectives

By the end of this section, you will be able to:

  • Solve applications using properties of triangles

Use the Pythagorean Theorem

  • Solve applications using rectangle properties

Be Prepared 3.12

Before you get started, take this readiness quiz.

Simplify: 1 2 ( 6 h ) . 1 2 ( 6 h ) . If you missed this problem, review Example 1.122 .

Be Prepared 3.13

The length of a rectangle is three less than the width. Let w represent the width. Write an expression for the length of the rectangle. If you missed this problem, review Example 1.26 .

Be Prepared 3.14

Solve: A = 1 2 b h A = 1 2 b h for b when A = 260 A = 260 and h = 52 . h = 52 . If you missed this problem, review Example 2.61 .

Be Prepared 3.15

Simplify: 144 . 144 . If you missed this problem, review Example 1.111 .

Solve Applications Using Properties of Triangles

In this section we will use some common geometry formulas. We will adapt our problem-solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve. In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications.

Solve Geometry Applications.

  • Step 1. Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information.
  • Step 2. Identify what we are looking for.
  • Step 3. Label what we are looking for by choosing a variable to represent it.
  • Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
  • Step 5. Solve the equation using good algebra techniques.
  • Step 6. Check the answer by substituting it back into the equation solved in step 5 and by making sure it makes sense in the context of the problem.
  • Step 7. Answer the question with a complete sentence.

We will start geometry applications by looking at the properties of triangles. Let’s review some basic facts about triangles. Triangles have three sides and three interior angles. Usually each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.

The plural of the word vertex is vertices . All triangles have three vertices . Triangles are named by their vertices: The triangle in Figure 3.4 is called △ A B C . △ A B C .

The three angles of a triangle are related in a special way. The sum of their measures is 180 ° . 180 ° . Note that we read m ∠ A m ∠ A as “the measure of angle A.” So in △ A B C △ A B C in Figure 3.4 ,

Because the perimeter of a figure is the length of its boundary, the perimeter of △ A B C △ A B C is the sum of the lengths of its three sides.

To find the area of a triangle, we need to know its base and height. The height is a line that connects the base to the opposite vertex and makes a 90 ° 90 ° angle with the base. We will draw △ A B C △ A B C again, and now show the height, h . See Figure 3.5 .

Triangle Properties

For △ A B C △ A B C

Angle measures:

  • The sum of the measures of the angles of a triangle is 180 ° . 180 ° .
  • The perimeter is the sum of the lengths of the sides of the triangle.
  • The area of a triangle is one-half the base times the height.

Example 3.34

The measures of two angles of a triangle are 55 and 82 degrees. Find the measure of the third angle.

Try It 3.67

The measures of two angles of a triangle are 31 and 128 degrees. Find the measure of the third angle.

Try It 3.68

The measures of two angles of a triangle are 49 and 75 degrees. Find the measure of the third angle.

Example 3.35

The perimeter of a triangular garden is 24 feet. The lengths of two sides are four feet and nine feet. How long is the third side?

Try It 3.69

The perimeter of a triangular garden is 48 feet. The lengths of two sides are 18 feet and 22 feet. How long is the third side?

Try It 3.70

The lengths of two sides of a triangular window are seven feet and five feet. The perimeter is 18 feet. How long is the third side?

Example 3.36

The area of a triangular church window is 90 square meters. The base of the window is 15 meters. What is the window’s height?

Try It 3.71

The area of a triangular painting is 126 square inches. The base is 18 inches. What is the height?

Try It 3.72

A triangular tent door has an area of 15 square feet. The height is five feet. What is the base?

The triangle properties we used so far apply to all triangles. Now we will look at one specific type of triangle—a right triangle. A right triangle has one 90 ° 90 ° angle, which we usually mark with a small square in the corner.

Right Triangle

A right triangle has one 90 ° 90 ° angle, which is often marked with a square at the vertex.

Example 3.37

One angle of a right triangle measures 28 ° . 28 ° . What is the measure of the third angle?

Try It 3.73

One angle of a right triangle measures 56 ° . 56 ° . What is the measure of the other small angle?

Try It 3.74

One angle of a right triangle measures 45 ° . 45 ° . What is the measure of the other small angle?

In the examples we have seen so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. We will wait to draw the figure until we write expressions for all the angles we are looking for.

Example 3.38

The measure of one angle of a right triangle is 20 degrees more than the measure of the smallest angle. Find the measures of all three angles.

Try It 3.75

The measure of one angle of a right triangle is 50° more than the measure of the smallest angle. Find the measures of all three angles.

Try It 3.76

The measure of one angle of a right triangle is 30° more than the measure of the smallest angle. Find the measures of all three angles.

We have learned how the measures of the angles of a triangle relate to each other. Now, we will learn how the lengths of the sides relate to each other. An important property that describes the relationship among the lengths of the three sides of a right triangle is called the Pythagorean Theorem . This theorem has been used around the world since ancient times. It is named after the Greek philosopher and mathematician, Pythagoras, who lived around 500 BC.

Before we state the Pythagorean Theorem, we need to introduce some terms for the sides of a triangle. Remember that a right triangle has a 90 ° 90 ° angle, marked with a small square in the corner. The side of the triangle opposite the 90 ° 90 ° angle is called the hypotenuse and each of the other sides are called legs .

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse. In symbols we say: in any right triangle, a 2 + b 2 = c 2 , a 2 + b 2 = c 2 , where a and b a and b are the lengths of the legs and c c is the length of the hypotenuse.

Writing the formula in every exercise and saying it aloud as you write it, may help you remember the Pythagorean Theorem.

The Pythagorean Theorem

In any right triangle, a 2 + b 2 = c 2 . a 2 + b 2 = c 2 .

where a and b are the lengths of the legs, c is the length of the hypotenuse.

To solve exercises that use the Pythagorean Theorem, we will need to find square roots. We have used the notation m m and the definition:

If m = n 2 , m = n 2 , then m = n , m = n , for n ≥ 0 . n ≥ 0 .

For example, we found that 25 25 is 5 because 25 = 5 2 . 25 = 5 2 .

Because the Pythagorean Theorem contains variables that are squared, to solve for the length of a side in a right triangle, we will have to use square roots.

Example 3.39

Use the Pythagorean Theorem to find the length of the hypotenuse shown below.

Try It 3.77

Use the Pythagorean Theorem to find the length of the hypotenuse in the triangle shown below.

Try It 3.78

Example 3.40.

Use the Pythagorean Theorem to find the length of the leg shown below.

Try It 3.79

Use the Pythagorean Theorem to find the length of the leg in the triangle shown below.

Try It 3.80

Example 3.41.

Kelvin is building a gazebo and wants to brace each corner by placing a 10 ″ 10 ″ piece of wood diagonally as shown above.

If he fastens the wood so that the ends of the brace are the same distance from the corner, what is the length of the legs of the right triangle formed? Approximate to the nearest tenth of an inch.

Try It 3.81

John puts the base of a 13-foot ladder five feet from the wall of his house as shown below. How far up the wall does the ladder reach?

Try It 3.82

Randy wants to attach a 17 foot string of lights to the top of the 15 foot mast of his sailboat, as shown below. How far from the base of the mast should he attach the end of the light string?

Solve Applications Using Rectangle Properties

You may already be familiar with the properties of rectangles. Rectangles have four sides and four right ( 90 ° ) ( 90 ° ) angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, L , and its adjacent side as the width, W .

The distance around this rectangle is L + W + L + W , L + W + L + W , or 2 L + 2 W . 2 L + 2 W . This is the perimeter , P , of the rectangle.

What about the area of a rectangle? Imagine a rectangular rug that is 2-feet long by 3-feet wide. Its area is 6 square feet. There are six squares in the figure.

The area is the length times the width.

The formula for the area of a rectangle is A = L W . A = L W .

Properties of Rectangles

Rectangles have four sides and four right ( 90 ° ) ( 90 ° ) angles.

The lengths of opposite sides are equal.

The perimeter of a rectangle is the sum of twice the length and twice the width.

The area of a rectangle is the product of the length and the width.

Example 3.42

The length of a rectangle is 32 meters and the width is 20 meters. What is the perimeter?

Try It 3.83

The length of a rectangle is 120 yards and the width is 50 yards. What is the perimeter?

Try It 3.84

The length of a rectangle is 62 feet and the width is 48 feet. What is the perimeter?

Example 3.43

The area of a rectangular room is 168 square feet. The length is 14 feet. What is the width?

Try It 3.85

The area of a rectangle is 598 square feet. The length is 23 feet. What is the width?

Try It 3.86

The width of a rectangle is 21 meters. The area is 609 square meters. What is the length?

Example 3.44

Find the length of a rectangle with perimeter 50 inches and width 10 inches.

Try It 3.87

Find the length of a rectangle with: perimeter 80 and width 25.

Try It 3.88

Find the length of a rectangle with: perimeter 30 and width 6.

We have solved problems where either the length or width was given, along with the perimeter or area; now we will learn how to solve problems in which the width is defined in terms of the length. We will wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

Example 3.45

The width of a rectangle is two feet less than the length. The perimeter is 52 feet. Find the length and width.

Try It 3.89

The width of a rectangle is seven meters less than the length. The perimeter is 58 meters. Find the length and width.

Try It 3.90

The length of a rectangle is eight feet more than the width. The perimeter is 60 feet. Find the length and width.

Example 3.46

The length of a rectangle is four centimeters more than twice the width. The perimeter is 32 centimeters. Find the length and width.

Try It 3.91

The length of a rectangle is eight more than twice the width. The perimeter is 64. Find the length and width.

Try It 3.92

The width of a rectangle is six less than twice the length. The perimeter is 18. Find the length and width.

Example 3.47

The perimeter of a rectangular swimming pool is 150 feet. The length is 15 feet more than the width. Find the length and width.

Try It 3.93

The perimeter of a rectangular swimming pool is 200 feet. The length is 40 feet more than the width. Find the length and width.

Try It 3.94

The length of a rectangular garden is 30 yards more than the width. The perimeter is 300 yards. Find the length and width.

Section 3.4 Exercises

Practice makes perfect.

Solving Applications Using Triangle Properties

In the following exercises, solve using triangle properties.

The measures of two angles of a triangle are 26 and 98 degrees. Find the measure of the third angle.

The measures of two angles of a triangle are 61 and 84 degrees. Find the measure of the third angle.

The measures of two angles of a triangle are 105 and 31 degrees. Find the measure of the third angle.

The measures of two angles of a triangle are 47 and 72 degrees. Find the measure of the third angle.

The perimeter of a triangular pool is 36 yards. The lengths of two sides are 10 yards and 15 yards. How long is the third side?

A triangular courtyard has perimeter 120 meters. The lengths of two sides are 30 meters and 50 meters. How long is the third side?

If a triangle has sides 6 feet and 9 feet and the perimeter is 23 feet, how long is the third side?

If a triangle has sides 14 centimeters and 18 centimeters and the perimeter is 49 centimeters, how long is the third side?

A triangular flag has base one foot and height 1.5 foot. What is its area?

A triangular window has base eight feet and height six feet. What is its area?

What is the base of a triangle with area 207 square inches and height 18 inches?

What is the height of a triangle with area 893 square inches and base 38 inches?

One angle of a right triangle measures 33 degrees. What is the measure of the other small angle?

One angle of a right triangle measures 51 degrees. What is the measure of the other small angle?

One angle of a right triangle measures 22.5 degrees. What is the measure of the other small angle?

One angle of a right triangle measures 36.5 degrees. What is the measure of the other small angle?

The perimeter of a triangle is 39 feet. One side of the triangle is one foot longer than the second side. The third side is two feet longer than the second side. Find the length of each side.

The perimeter of a triangle is 35 feet. One side of the triangle is five feet longer than the second side. The third side is three feet longer than the second side. Find the length of each side.

One side of a triangle is twice the shortest side. The third side is five feet more than the shortest side. The perimeter is 17 feet. Find the lengths of all three sides.

One side of a triangle is three times the shortest side. The third side is three feet more than the shortest side. The perimeter is 13 feet. Find the lengths of all three sides.

The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.

The measure of the smallest angle of a right triangle is 20° less than the measure of the next larger angle. Find the measures of all three angles.

The angles in a triangle are such that one angle is twice the smallest angle, while the third angle is three times as large as the smallest angle. Find the measures of all three angles.

The angles in a triangle are such that one angle is 20° more than the smallest angle, while the third angle is three times as large as the smallest angle. Find the measures of all three angles.

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

In the following exercises, use the Pythagorean Theorem to find the length of the leg. Round to the nearest tenth, if necessary.

In the following exercises, solve using the Pythagorean Theorem. Approximate to the nearest tenth, if necessary.

A 13-foot string of lights will be attached to the top of a 12-foot pole for a holiday display, as shown below. How far from the base of the pole should the end of the string of lights be anchored?

Pam wants to put a banner across her garage door, as shown below, to congratulate her son for his college graduation. The garage door is 12 feet high and 16 feet wide. How long should the banner be to fit the garage door?

Chi is planning to put a path of paving stones through her flower garden, as shown below. The flower garden is a square with side 10 feet. What will the length of the path be?

Brian borrowed a 20 foot extension ladder to use when he paints his house. If he sets the base of the ladder 6 feet from the house, as shown below, how far up will the top of the ladder reach?

In the following exercises, solve using rectangle properties.

The length of a rectangle is 85 feet and the width is 45 feet. What is the perimeter?

The length of a rectangle is 26 inches and the width is 58 inches. What is the perimeter?

A rectangular room is 15 feet wide by 14 feet long. What is its perimeter?

A driveway is in the shape of a rectangle 20 feet wide by 35 feet long. What is its perimeter?

The area of a rectangle is 414 square meters. The length is 18 meters. What is the width?

The area of a rectangle is 782 square centimeters. The width is 17 centimeters. What is the length?

The width of a rectangular window is 24 inches. The area is 624 square inches. What is the length?

The length of a rectangular poster is 28 inches. The area is 1316 square inches. What is the width?

Find the length of a rectangle with perimeter 124 and width 38.

Find the width of a rectangle with perimeter 92 and length 19.

Find the width of a rectangle with perimeter 16.2 and length 3.2.

Find the length of a rectangle with perimeter 20.2 and width 7.8.

The length of a rectangle is nine inches more than the width. The perimeter is 46 inches. Find the length and the width.

The width of a rectangle is eight inches more than the length. The perimeter is 52 inches. Find the length and the width.

The perimeter of a rectangle is 58 meters. The width of the rectangle is five meters less than the length. Find the length and the width of the rectangle.

The perimeter of a rectangle is 62 feet. The width is seven feet less than the length. Find the length and the width.

The width of the rectangle is 0.7 meters less than the length. The perimeter of a rectangle is 52.6 meters. Find the dimensions of the rectangle.

The length of the rectangle is 1.1 meters less than the width. The perimeter of a rectangle is 49.4 meters. Find the dimensions of the rectangle.

The perimeter of a rectangle is 150 feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.

The length of a rectangle is three times the width. The perimeter of the rectangle is 72 feet. Find the length and width of the rectangle.

The length of a rectangle is three meters less than twice the width. The perimeter of the rectangle is 36 meters. Find the dimensions of the rectangle.

The length of a rectangle is five inches more than twice the width. The perimeter is 34 inches. Find the length and width.

The perimeter of a rectangular field is 560 yards. The length is 40 yards more than the width. Find the length and width of the field.

The perimeter of a rectangular atrium is 160 feet. The length is 16 feet more than the width. Find the length and width of the atrium.

A rectangular parking lot has perimeter 250 feet. The length is five feet more than twice the width. Find the length and width of the parking lot.

A rectangular rug has perimeter 240 inches. The length is 12 inches more than twice the width. Find the length and width of the rug.

Everyday Math

Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are six feet, eight feet and 10 feet. How many feet of fencing will she need to enclose her flowerbed?

Jose just removed the children’s playset from his back yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a 50 foot roll of fence in his garage that he plans to use. To fit in the backyard, the width of the garden must be 10 feet. How long can he make the other length?

Writing Exercises

If you need to put tile on your kitchen floor, do you need to know the perimeter or the area of the kitchen? Explain your reasoning.

If you need to put a fence around your backyard, do you need to know the perimeter or the area of the backyard? Explain your reasoning.

Look at the two figures below.

  • ⓐ Which figure looks like it has the larger area?
  • ⓑ Which looks like it has the larger perimeter?
  • ⓒ Now calculate the area and perimeter of each figure.
  • ⓓ Which has the larger area?
  • ⓔ Which has the larger perimeter?

Write a geometry word problem that relates to your life experience, then solve it and explain all your steps.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
  • Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
  • Publisher/website: OpenStax
  • Book title: Elementary Algebra 2e
  • Publication date: Apr 22, 2020
  • Location: Houston, Texas
  • Book URL: https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
  • Section URL: https://openstax.org/books/elementary-algebra-2e/pages/3-4-solve-geometry-applications-triangles-rectangles-and-the-pythagorean-theorem

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3.4: Triangles, Rectangles, and the Pythagorean Theorem

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Learning Objectives

By the end of this section, you will be able to:

  • Solve applications using properties of triangles

Use the Pythagorean Theorem

  • Solve applications using rectangle properties

Be prepared

Before you get started, take this readiness quiz.

  • Simplify: \(12(6h)\). If you missed this problem, review Exercise 1.10.1 .
  • The length of a rectangle is three less than the width. Let w represent the width. Write an expression for the length of the rectangle. If you missed this problem, review Exercise 1.3.43 .
  • Solve: \(A=\frac{1}{2}bh\) for b when A=260 and h=52. If you missed this problem, review Exercise 2.6.10 .
  • Simplify: \(\sqrt{144}\). If you missed this problem, review Exercise 1.9.10 .

Solve Applications Using Properties of Triangles

In this section we will use some common geometry formulas. We will adapt our problem-solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve. In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications.

SOLVE GEOMETRY APPLICATIONS

  • Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information.
  • Identify what we are looking for.
  • Label what we are looking for by choosing a variable to represent it.
  • Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
  • Solve the equation using good algebra techniques.
  • Check the answer by substituting it back into the equation solved in step 5 and by making sure it makes sense in the context of the problem.
  • Answer the question with a complete sentence.

We will start geometry applications by looking at the properties of triangles. Let’s review some basic facts about triangles. Triangles have three sides and three interior angles. Usually each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.

The plural of the word vertex is vertices . All triangles have three vertices . Triangles are named by their vertices: The triangle in Figure \(\PageIndex{1}\) is called \(\triangle{ABC}\).

A triangle with vertices A, B, and C. The sides opposite these vertices are marked a, b, and c, respectively.

The three angles of a triangle are related in a special way. The sum of their measures is \(180^{\circ}\). Note that we read \(m\angle{A}\) as “the measure of angle A.” So in \(\triangle{ABC}\) in Figure \(\PageIndex{1}\).

\[m \angle A+m \angle B+m \angle C=180^{\circ} \nonumber\]

Because the perimeter of a figure is the length of its boundary, the perimeter of \(\triangle{ABC}\) is the sum of the lengths of its three sides.

\[P = a + b + c \nonumber\]

To find the area of a triangle, we need to know its base and height. The height is a line that connects the base to the opposite vertex and makes a \(90^\circ\) angle with the base. We will draw \(\triangle{ABC}\) again, and now show the height, \(h\). See Figure \(\PageIndex{2}\).

A triangle with vertices A, B, and C. The sides opposite these vertices are marked a, b, and c, respectively. The side b is parallel to the bottom of the page, and it has a dashed line drawn from vertex B to it. This line is marked h and makes a right angle with side b.

TRIANGLE PROPERTIES

A triangle with vertices A, B, and C. The sides opposite these vertices are marked a, b, and c, respectively. The side b is parallel to the bottom of the page, and it has a dashed line drawn from vertex B to it. This line is marked h and makes a right angle with side b.

For \(\triangle{ABC}\)

Angle measures:

\[m \angle A+m \angle B+m \angle C=180^{\circ}\]

  • The sum of the measures of the angles of a triangle is 180°.

\[P = a + b + c\]

  • The perimeter is the sum of the lengths of the sides of the triangle.

\(A = \frac{1}{2}bh, b = \text{ base }, h = \text{ height }\)

  • The area of a triangle is one-half the base times the height.

Example \(\PageIndex{1}\)

The measures of two angles of a triangle are 55 and 82 degrees. Find the measure of the third angle.

Try It \(\PageIndex{1}\)

The measures of two angles of a triangle are 31 and 128 degrees. Find the measure of the third angle.

Try It \(\PageIndex{2}\)

The measures of two angles of a triangle are 49 and 75 degrees. Find the measure of the third angle.

Example \(\PageIndex{2}\)

The perimeter of a triangular garden is 24 feet. The lengths of two sides are four feet and nine feet. How long is the third side?

Try It \(\PageIndex{3}\)

The perimeter of a triangular garden is 48 feet. The lengths of two sides are 18 feet and 22 feet. How long is the third side?

Try It \(\PageIndex{4}\)

The lengths of two sides of a triangular window are seven feet and five feet. The perimeter is 18 feet. How long is the third side?

Example \(\PageIndex{3}\)

The area of a triangular church window is 90 square meters. The base of the window is 15 meters. What is the window’s height?

Try It \(\PageIndex{5}\)

The area of a triangular painting is 126 square inches. The base is 18 inches. What is the height?

Try It \(\PageIndex{6}\)

A triangular tent door has area 15 square feet. The height is five feet. What is the base?

The triangle properties we used so far apply to all triangles. Now we will look at one specific type of triangle—a right triangle. A right triangle has one 90° angle, which we usually mark with a small square in the corner.

A right triangle with the largest angle marked 90 degrees.

Definition: RIGHT TRIANGLE

A right triangle has one 90° angle, which is often marked with a square at the vertex.

Example \(\PageIndex{4}\)

One angle of a right triangle measures 28°. What is the measure of the third angle?

Try It \(\PageIndex{7}\)

One angle of a right triangle measures 56°. What is the measure of the other small angle?

Try It \(\PageIndex{8}\)

One angle of a right triangle measures 45°. What is the measure of the other small angle?

In the examples we have seen so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. We will wait to draw the figure until we write expressions for all the angles we are looking for.

Example \(\PageIndex{5}\)

The measure of one angle of a right triangle is 20 degrees more than the measure of the smallest angle. Find the measures of all three angles.

Try It \(\PageIndex{9}\)

The measure of one angle of a right triangle is 50° more than the measure of the smallest angle. Find the measures of all three angles.

20°,70°,90°

Try It \(\PageIndex{10}\)

The measure of one angle of a right triangle is 30° more than the measure of the smallest angle. Find the measures of all three angles.

30°,60°,90°

We have learned how the measures of the angles of a triangle relate to each other. Now, we will learn how the lengths of the sides relate to each other. An important property that describes the relationship among the lengths of the three sides of a right triangle is called the Pythagorean Theorem . This theorem has been used around the world since ancient times. It is named after the Greek philosopher and mathematician, Pythagoras, who lived around 500 BC.

Before we state the Pythagorean Theorem, we need to introduce some terms for the sides of a triangle. Remember that a right triangle has a 90° angle, marked with a small square in the corner. The side of the triangle opposite the 90°90° angle is called the hypotenuse and each of the other sides are called legs .

Three right triangles with different orientations. The right angles are marked with two small lines that make a small square with the angle. Opposite these angles, hypotenuse is written. The other sides are marked “leg.”

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse. In symbols we say: in any right triangle, \(a^{2}+b^{2}=c^{2}\), where a and b are the lengths of the legs and cc is the length of the hypotenuse.

Writing the formula in every exercise and saying it aloud as you write it, may help you remember the Pythagorean Theorem.

THE PYTHAGOREAN THEOREM

In any right triangle, where \(a\) and \(b\) are the lengths of the legs, \(c\) is the length of the hypotenuse.

A right triangle with sides marked a, b, and c. The side marked c is the hypotenuse.

\[a^{2}+b^{2}=c^{2} \label{Ptheorem}\]

To solve exercises that use the Pythagorean Theorem (Equation \ref{Ptheorem}), we will need to find square roots. We have used the notation \(\sqrt{m}\) and the definition:

If \(m = n^{2}\), then \(\sqrt{m} = n\), for \(n\geq 0\).

For example, we found that \(\sqrt{25}\) is 5 because \(25=5^{2}\).

Because the Pythagorean Theorem contains variables that are squared, to solve for the length of a side in a right triangle, we will have to use square roots.

Example \(\PageIndex{6}\)

Use the Pythagorean Theorem to find the length of the hypotenuse shown below.

A right triangle with legs marked 3 and 4.

Try It \(\PageIndex{11}\)

Use the Pythagorean Theorem to find the length of the hypotenuse in the triangle shown below.

A right triangle with legs marked 6 and 8. The hypotenuse is marked c.

Try It \(\PageIndex{12}\)

No Alt Text

Example \(\PageIndex{7}\)

Use the Pythagorean Theorem to find the length of the leg shown below.

A right angle with one leg marked 5. The hypotenuse is labeled 13.

Try It \(\PageIndex{13}\)

Use the Pythagorean Theorem to find the length of the leg in the triangle shown below.

A right triangle with legs marked b and 15. The hypotenuse is marked 17.

Try It \(\PageIndex{14}\)

A right triangle with legs marked b and 9. The hypotenuse is marked 15.

Example \(\PageIndex{8}\)

A gazebo is shown. In one of its corners, a triangle is made with the wood. The hypotenuse is marked 10 inches, and one of the legs is marked x

Kelvin is building a gazebo and wants to brace each corner by placing a 10″ piece of wood diagonally as shown above.

If he fastens the wood so that the ends of the brace are the same distance from the corner, what is the length of the legs of the right triangle formed? Approximate to the nearest tenth of an inch.

\(\begin{array} {ll} {\textbf{Step 1. }\text{Read the problem.}} &{} \\\\ {\textbf{Step 2. }\text{Identify what we are looking for.}} &{\text{the distance from the corner that the}} \\ {} &{\text{bracket should be attached}} \\ \\{\textbf{Step 3. }\text{Name. Choose a variable to represent it.}} &{\text{Let x = distance from the corner.}} \\ {\textbf{Step 4.} \text{Translate}} &{} \\ {\text{Write the appropriate formula and substitute.}} &{a^{2} + b^{2} = c^{2}} \\ {} &{x^{2} + x^{2} = 10^{2}} \\ \\ {\textbf{Step 5. Solve the equation.}} &{} \\ {} &{2x^{2} = 100} \\ {\text{Isolate the variable.}} &{x^{2} = 50} \\ {\text{Simplify. Approximate to the nearest tenth.}} &{x \approx 7.1}\\\\ {\textbf{Step 6. }\text{Check.}} &{}\\ {a^{2} + b^{2} = c^{2}} &{} \\ {(7.1)^{2} + (7.1)^{2} \approx 10^{2} \text{ Yes.}} &{} \\\\ {\textbf{Step 7. Answer the question.}} &{\text{Kelven should fasten each piece of}} \\ {} &{\text{wood approximately 7.1'' from the corner.}} \end{array}\)

Try It \(\PageIndex{15}\)

John puts the base of a 13-foot ladder five feet from the wall of his house as shown below. How far up the wall does the ladder reach?

A house is shown with a ladder leaning against it. The ladder is marked 13’, and the distance from the house to the base of the ladder is marked 5’.

Try It \(\PageIndex{16}\)

Randy wants to attach a 17 foot string of lights to the top of the 15 foot mast of his sailboat, as shown below. How far from the base of the mast should he attach the end of the light string?

A sailboat is shown with a 15’ mast (the straight tall part). From the top of the mast, a series of colored dots stretches down to the back of the boat and is marked 17’.

Solve Applications Using Rectangle Properties

You may already be familiar with the properties of rectangles. Rectangles have four sides and four right (90°) angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, \(L\), and its adjacent side as the width, \(W\).

A rectangle with sides marked W and L.

The distance around this rectangle is \(L+W+L+W\), or \(2L+2W\). This is the perimeter , \(P\), of the rectangle.

\[P=2L+2W\]

What about the area of a rectangle? Imagine a rectangular rug that is 2-feet long by 3-feet wide. Its area is 6 square feet. There are six squares in the figure.

A rectangles composed of 6 squares that is three high and two wide. The height is marked 3 and the width is marked 2.

\[\begin{array} {l} {A=6} \\ {A=2\cdot3} \\ {A=L\cdot W} \end{array}\]

The area is the length times the width. The formula for the area of a rectangle is

PROPERTIES OF RECTANGLES

  • Rectangles have four sides and four right (90°) angles.
  • The lengths of opposite sides are equal.

The perimeter of a rectangle is the sum of twice the length and twice the width.

The area of a rectangle is the product of the length and the width.

\[A=L·W\]

Example \(\PageIndex{9}\)

The length of a rectangle is 32 meters and the width is 20 meters. What is the perimeter?

Try It \(\PageIndex{17}\)

The length of a rectangle is 120 yards and the width is 50 yards. What is the perimeter?

Try It \(\PageIndex{18}\)

The length of a rectangle is 62 feet and the width is 48 feet. What is the perimeter?

Example \(\PageIndex{10}\)

The area of a rectangular room is 168 square feet. The length is 14 feet. What is the width?

Try It \(\PageIndex{19}\)

The area of a rectangle is 598 square feet. The length is 23 feet. What is the width?

Try It \(\PageIndex{20}\)

The width of a rectangle is 21 meters. The area is 609 square meters. What is the length?

Example \(\PageIndex{11}\)

Find the length of a rectangle with perimeter 50 inches and width 10 inches.

Try It \(\PageIndex{21}\)

Find the length of a rectangle with: perimeter 80 and width 25.

Try It \(\PageIndex{22}\)

Find the length of a rectangle with: perimeter 30 and width 6.

We have solved problems where either the length or width was given, along with the perimeter or area; now we will learn how to solve problems in which the width is defined in terms of the length. We will wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

Example \(\PageIndex{12}\)

The width of a rectangle is two feet less than the length. The perimeter is 52 feet. Find the length and width.

Try It \(\PageIndex{23}\)

The width of a rectangle is seven meters less than the length. The perimeter is 58 meters. Find the length and width.

18 meters, 11 meters

Try It \(\PageIndex{24}\)

The length of a rectangle is eight feet more than the width. The perimeter is 60 feet. Find the length and width.

19 feet, 11 feet

Example \(\PageIndex{13}\)

The length of a rectangle is four centimeters more than twice the width. The perimeter is 32 centimeters. Find the length and width.

Try It \(\PageIndex{25}\)

The length of a rectangle is eight more than twice the width. The perimeter is 64. Find the length and width.

Try It \(\PageIndex{26}\)

The width of a rectangle is six less than twice the length. The perimeter is 18. Find the length and width.

Example \(\PageIndex{14}\)

The perimeter of a rectangular swimming pool is 150 feet. The length is 15 feet more than the width. Find the length and width.

Try It \(\PageIndex{27}\)

The perimeter of a rectangular swimming pool is 200 feet. The length is 40 feet more than the width. Find the length and width.

70 feet, 30 feet

Try It \(\PageIndex{28}\)

The length of a rectangular garden is 30 yards more than the width. The perimeter is 300 yards. Find the length and width.

90 yards, 60 yards

Key Concepts

  • Read the problem and make all the words and ideas are understood. Draw the figure and label it with the given information.
  • Name what we are looking for by choosing a variable to represent it.
  • Check the answer in the problem and make sure it makes sense.
  • \(m\angle{A}+m\angle{B}+m\angle{C}=180\)
  • \(P=a+b+c\)
  • \(A=\frac{1}{2}bh\), b=base,h=height
  • The Pythagorean Theorem In any right triangle, \(a^{2} + b^{2} = c^{2}\) where \(c\) is the length of the hypotenuse and \(a\) and \(b\) are the lengths of the legs.
  • The perimeter of a rectangle is the sum of twice the length and twice the width: \(P=2L+2W\).
  • The area of a rectangle is the length times the width: \(A=LW\).

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  1. Perimeter of a Triangle

    9 + 9 + 12 = 30 9 + 9 + 12 = 30 2 Write the final answer with the correct units. The side lengths are measured in feet, so the total perimeter is in feet. The perimeter of the triangle is 30 30 feet. Example 2: perimeter of a right triangle What is the perimeter of the triangle? Add all the side lengths. Show step

  2. Perimeter of a Triangle. Calculator

    In real-life problems, a perimeter of a triangle may be useful in making a fence around the triangular parcel, tying up a triangular box with ribbon, or estimating the lace needed for binding a triangular pennant. However, we guess that you will probably use it in your Maths class ;) How to find the perimeter of the triangle?

  3. 3 Simple Ways to Find the Perimeter of a Triangle

    1 Remember the formula for finding the perimeter of a triangle. For a triangle with sides a, b and c, the perimeter P is defined as: P = a + b + c. [2] What this formula means in simpler terms is that to find the perimeter of a triangle, you just add together the lengths of each of its 3 sides. 2

  4. How to find the perimeter of a right triangle

    Now, plug in all three values into the equation to find the perimeter. Use a calculator and round to decimal places. There are three primary methods used to find the perimeter of a right triangle. When side lengths are given, add them together. Solve for a missing side using the Pythagorean theorem.

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    Welcome to How to Find the Perimeter of a Triangle with Mr. J! Need help with finding the perimeter of a triangle? You're in the right place!Whether you're j...

  7. Perimeter of a Triangle- Definition, Formula and Examples

    Solved Examples Let us consider some of the examples on the perimeter of a triangle: Example 1: Find the perimeter of a polygon whose sides are 5 cm, 4 cm and 2 cm. Solution: Let, a = 5 cm b = 4 cm c = 2 cm Perimeter = Sum of all sides = a + b + c = 5 + 4 + 2 = 11 Therefore, the answer is 11 cm.

  8. Perimeter of a Triangle: Definition, Formula, Examples, FAQs

    Step 1: Note down the lengths of all three sides of the given triangle. Ensure that the lengths are in the same unit. Step 2: Add the lengths of the three sides. Step 3: The sum represents the perimeter of the given triangle. Assign the same unit to the perimeter as the length of the sides. Perimeter of an Equilateral Triangle

  9. How to Find the Perimeter of a Triangle (Formula & Video)

    Well done! You used algebra to solve a perimeter problem! Lesson summary. Now that you have worked your way through the lesson, you are able to define perimeter, recognize the types of triangles, recall and explain a method of finding the perimeter of triangles by adding the lengths of their sides, and, given perimeter, solve for lengths of sides of a triangle using algebra.

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    Example Problem 2. An equilateral triangle has a perimeter measured to be 81 centimeters. What are the side lengths? Solution: Since an equilateral triangle has three equal sides, a = b = c. We can substitute a + b + c with 3a. Plugging what we know into the perimeter formula, we get: P = a + b + c 81 = 3a a = 27, b = 27, c = 27

  11. Perimeter of a Triangle

    That means our perimeter is simply. A right triangle has two short sides with lengths 5 and 12. What is the perimeter of the triangle? represent the two shorter sides of the triangle, and. The perimeter of the triangle is 30. Solve for the perimeter of the triangle. Tom's, Bob's, and Fred's houses are in a triangle. away from Bob's house.

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    Solved Examples Frequently Asked Questions What is a polygon? Figures can be open or closed, which may be made up of straight lines and curved lines, or made up of only curves, or only lines. If a figure is closed and made up of only line segments, it is called a polygon. What is a triangle?

  13. Perimeter of a Triangle Worksheets

    Calculate the perimeter by plugging in the side lengths given as fractions or mixed numbers in the formula P = a + b + c, where a, b, c are the three sides of the triangle. Convert to mixed numbers if required. Download the set Perimeter of a Triangle | Fractions - Type 2

  14. Perimeter of a triangle

    The formula for the perimeter of a triangle is: Perimeter = a + b + c where a, b, and c are the lengths of the three sides of the triangle. Special triangles Isosceles triangles, equilateral triangles, and right triangles have a number of relationships that allow us to find their perimeters without necessarily knowing all of their side lengths.

  15. Perimeter of a Triangle Formula

    The perimeter of the triangle = Sum of the lengths of three sides. Let us consider some of the examples on perimeter of a triangle: 1. Find the perimeter of a triangle having sides 3 cm, 8 cm and 6 cm. Solution: Perimeter of a triangle. = Sum of all the three sides. = AB + BC + AC. = 3 cm + 8 cm + 6 cm.

  16. Perimeter: Problems with Solutions

    By Catalin David Problem 1 What is the perimeter of triangle ABC? Problem 2 The red triangle is equilateral with a side of 23 centimetres. Its perimeter is cm. Problem 3 An isosceles triangle has a perimetеr of 37 centimetres and its base has a length of 9 centimetres. Each of the other two sides has a length of cm. Problem 4

  17. Area and Perimeter of a Triangle

    Problem-Solving Examples. We can now proceed to solve sample problems to apply what we have learned so far. Each problem tackles different formulas discussed and gives us a challenge on how to solve through the information given to us. Perimeter of a Triangle. Sample Problem 1:

  18. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the

    Solve Geometry Applications. Step 1. Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information. Step 2. Identify what we are looking for.; Step 3. Label what we are looking for by choosing a variable to represent it.; Step 4. Translate into an equation by writing the appropriate formula or model for the situation.

  19. Area and Perimeter of Triangles: Meaning, Formulas, Problems

    Written By SHWETHA B.R Last Modified 25-01-2023 Area and Perimeter of Triangles: Meaning, Formulas, Problems Area and Perimeter of Triangles: Perimeter is the total length of the three sides of any triangle. The area of a triangle is the region or surface bounded by the shape of a Triangle.

  20. 3.4: Triangles, Rectangles, and the Pythagorean Theorem

    Perimeter: P = a + b + c. Area: A = 1 2bh, b=base,h=height. A right triangle has one 90° angle. The Pythagorean Theorem In any right triangle, a2 + b2 = c2 where c is the length of the hypotenuse and a and b are the lengths of the legs. Properties of Rectangles. Rectangles have four sides and four right (90°) angles.

  21. Perimeter Word Problems: Examples

    Show Step-by-step Solutions Word Problems - Perimeter of a triangle Example: Patrick's bike ride follows a triangular path; two legs are equal, the third is 8 miles longer than each of the other legs. If Patrick rides 30 miles total, what is the length of the longest leg? Word Problem Involving Perimeter of a Triangle Example:

  22. CK12-Foundation

    To find the perimeter, we need to find the longest side of the obtuse triangle. If we used the black lines in the picture, we would see that the longest side is also the hypotenuse of the right triangle with legs 4 and 10. 42 + 102 = c2 16 + 100 = c2 c = √116 ≈ 10.77. The perimeter is 7 + 5 + 10.77 ≈ 22.77 units.

  23. PDF Year 3 Calculate Perimeter Reasoning and Problem Solving

    National Curriculum Objectives: Mathematics Year 3: (3M7) Measure the perimeter of simple 2-D shapes Differentiation: Questions 1, 4 and 7 (Problem Solving) Developing Investigate what regular shape may have been drawn and the length of each of its sides, using the given clues. Number of sides each shape has is given.

  24. The Mysterious Math of Billiards Tables

    A Triangle Inequality. Billiards in triangles, which do not have the nice right-angled geometry of rectangles, is more complicated. As you might remember from high school geometry, there are several kinds of triangles: acute triangles, where all three internal angles are less than 90 degrees; right triangles, which have a 90-degree angle; and obtuse triangles, which have one angle that is more ...

  25. Math Message Boards FAQ & Community Help

    Art of Problem Solving AoPS Online. Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12 ...