3.6 Calculating the Length of a Side in a Right Triangle

In this lesson, we will go over how to find a missing side length in a right triangle. We will cover how to find a missing side length using the Pythagorean theorem and how to find a missing side length using the trigonometric ratios. We will also distinguish when we can use each of these methods.

The Pythagorean Theorem

The Pythagorean theorem is a special property of right triangles. It is named after the Greek philosopher and mathematician Pythagoras, who lived around 500 BCE. Although Pythagoras is credited with the theorem, there is evidence that many ancient civilizations used the theorem prior to his time. 

Remember that a right triangle has a 90° angle, which we usually mark with a small square in the corner. The side of the triangle opposite the 90° angle is called the hypotenuse. When using the Pythagorean theorem, the other two sides are called the legs. The diagrams below show three right angle triangles with their sides labelled as mentioned above.

Three right triangles are shown. Each has a box representing the right angle. The first one has the right angle in the lower left corner, the next in the upper left corner, and the last one at the top. The two sides touching the right angle are labeled “leg” in each triangle. The sides across from the right angles are labeled “hypotenuse.”

 

 

 

The Pythagorean theorem can be used to find a side length of a right triangle given that you know the other two side lengths. The Pythagorean theorem shows 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 two legs equals the square of the hypotenuse.

Table 3.6.1
The Pythagorean Theorem
A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.

Given the right triangle \Delta ABC where c is the length of the hypotenuse a and b are the lengths of the legs,

{a}^{2}+{b}^{2}={c}^{2}
Example 1

Use the Pythagorean theorem to find the length of the hypotenuse.

Right triangle with legs labeled as 3 and 4.

Table 3.6.2
Steps Solution
Label the side lengths a, b, and c. Note that a and b are interchangeable. .a=\text{3}

b=\text{4}

c=\text{hypotenuse}
Substitute the values into the formula. {a}^{2}+{b}^{2}={c}^{2}

{3}^{2}+{4}^{2}={c}^{2}
Solve the equation. 9+16={c}^{2}

25={c}^{2}

\(\sqrt{25} = \sqrt{{c}^{2}}

5=c
Check your answer. {3}^{2}+{4}^{2}={5}^{2}

9+16=25

25=25
Practice 1

Find the length of the hypotenuse.

a)

A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.

 

 b)

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as c. One of the sides touching the right angle is labeled as 15, the other is labeled “8”.
Example 2

Use the Pythagorean theorem to find the length of the longer leg.

Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.

Table 3.6.3
Steps Solution
Label the side lengths a, b, and c. .\(a=\text{5}\)

b=\text{leg}

c=\text{13}
Substitute the values into the formula. {a}^{2}+{b}^{2}={c}^{2}

{5}^{2}+{b}^{2}={13}^{2}
Solve the equation. 25+{b}^{2}=169

{b}^{2}=144

\(\sqrt{{b}^{2}} = \sqrt{144}

b=12
Check your answer. {5}^{2}+{12}^{2}={13}^{2}

25+144=169

169=169
Practice 2

Find the length of side b.

a)

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15, the other is labeled “b”.

 b)

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 15. One of the sides touching the right angle is labeled as 9, the other is labeled “b”.

 

 

Trigonometric Ratios

The three primary trigonometric ratios discussed in the previous lesson can also be used to find the missing side lengths in a right triangle. Given one acute angle and the length of one side, the trigonometric ratios can be used to find unknown side lengths in a right triangle.

Example 3

Find side y. Round your final answer to two decimal places.

Right triangle XYZ. X is the right angle. Y is 35 degrees. x is 14.

Table 3.6.4
Steps Solution
Identify which side you are looking for and which side you have been given. We are looking for side y, which is the opposite side. We have been given side x = 14, which is the hypotenuse.
Identify which trigonometric ratio you will be using (sin, cos, or tan). We will use the sin ratio because we are working with the opposite and hypotenuse sides of the triangle.
Set up the trigonometric ratio. sin θ° =  \frac{opp}{hyp}

sin 35° =  \frac{y}{14}
Solve the ratio. 14 sin 35° = y

8.03 cm = y
Practice 3

Find the indicated side. Round your final answers to one decimal place.

a)

 

 

 

 

 b) Given that a = 10, find c.

A right triangle ABC. B is 90 degrees. C is 27 degrees.
Example 4

Find the hypotenuse. Round your final answer to one decimal place.

Right triangle PRS P is the right angle. S is 32 degrees. r is 4.

Table 3.6.5
Steps Solution
Identify which side you are looking for and which side you have been given. We are looking for side p, which is the hypotenuse. We have been given side r = 4, which is the adjacent side.
Identify which trigonometric ratio you will be using (sin, cos, or tan). We will use the cos ratio because we are working with the adjacent and hypotenuse sides of the triangle.
Set up the trigonometric ratio. cos θ° =  \frac{adj}{hyp}

cos 32° =  \frac{4}{p}
Solve the ratio. 0.8480 = \frac{4}{p}

p = 4.7
Practice 4

Find the hypotenuse. Round your final answers to one decimal place.

a)

Right triangle PRS. P is the right angle. R is 72 degrees. s is 7.

 b)

Right triangle PRS. P is the right angle. S is 38 degrees. s is 4.
Practice 5

Find the missing sides. Round your final answers to one decimal place.

Right triangle CAB. C is the right angle. A is 51 degrees. c is 26.

Homework

  1. Find the length of the hypotenuse.

    a)
    A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 9, the other as 12.
    		 b)
    A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12.
    c)
    A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 15, the other as 20.
    		 d)
    A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12.

  2. Find the length of the missing side. Round to the nearest tenth, if necessary.

    a)
    A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 10. One of the sides touching the right angle is labeled as 6.    		 b)
    A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 5.
    c)
    A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 8.    		 d)

    A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15.

  3. For the given triangles, find the missing side. Round your answers to one decimal place.

    a) Find the hypotenuse.
    A right triangle ABC. B is 90 degrees. C is 27 degrees. c is 9.    		 b) Given that a = 6, find b.
    A right triangle ABC. B is 90 degrees. C is 27 degrees.
    c) Find side c.

    A right triangle CDB. D is 90 degrees. C is 39 degrees. d is 19.

    		 d) Find side c.

    A right triangle CDB. D is 90 degrees. B is 57 degrees. d is 40.
    e) Given that a = 6, find c.

    A right triangle ABC. B is 90 degrees. C is 27 degrees.

    		 f) Find side a.

    A right triangle ABC. B is 90 degrees. C is 27 degrees. c is 9.

  4. For the given triangles, find all the missing sides. Round your answers to one decimal place.

    a)
    A right triangle ZXY. X is 90 degrees. Y is 67 degrees. x is 21.    		 b)
    A right triangle ZXY. X is 90 degrees. Z is 73 degrees. x is 16.

  5. In triangle ABC, \angleABC = 90°, \angleBAC = 15°, and side AB = 12 cm. Find the length of side BC. Round to the nearest hundredth.

     

     

     

  6. In \triangleMNO, \angleMNO = 90°, \angleMON = 48°, and side NM = 8 in. Find the length of side OM. Round to the nearest hundredth.

     

     

     

  7. Angela is building a gazebo and wants to brace each corner by placing a 10-inch wooden bracket diagonally as shown. How far below the corner should she fasten the bracket if she wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch. Use the diagram below to help you answer the question.

    A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.

  8. An airplane takes off from the ground at an angle of 25°. If the airplane travels 200 kilometres, how high above the ground is it? Round to the nearest tenth.

     

     

     

  9. Nadia puts the base of a 13-foot ladder 5 feet from the wall of his house. How far up the wall does the ladder reach?

     

     

     

  10. Can trigonometric ratios be used to find the length of the following sides in the given triangles? Explain why or why not for each triangle.

    a) side e
    A right triangle FEG. E is 90 degrees. F is theta degrees. g is 5.    		 b) side c
    A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.
    c) side a
    A right triangle ACB. C is 90 degrees. B is 74 degrees. b is 10.    		 d) side a
    A triangle ABC. A is 82 degrees. B is 55 degrees. C is x.

Answers

1.

a) 15 b) 20 c) 25 d) 13

2.

a) 8 b) 12 c) 10.2 d) 16.2

3.

a) 19.8 b) 6.7 c) 12 d) 21.8 e) 3.1 f) 17.7

4.

a) y = 19.3, z = 8.2 b) y = 4.7, z = 15.3
5. 3.22 cm 6. 10.77 7. 7.1 in 8. 84.5 km 9. 12 ft

10.

a) No. We need another angle measure. b) No. We need another angle measure.
c) Yes. We have a side length and an angle measure. d) No. This is not a right angle triangle.

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resources:

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

License

Icon for the Creative Commons Attribution-NonCommercial 4.0 International License

Math 10C Workbook Copyright © 2026 by Leanne Thompson is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

Share This Book