How to Find the Hypotenuse of a Right Triangle Using the Pythagorean Theorem

How to Find the Hypotenuse of a Right Triangle Using the Pythagorean Theorem
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  • 0:05 A Right Triangle
  • 1:25 The Pythagorean Theorem
  • 3:07 Example
  • 4:09 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will be able to use the Pythagorean Theorem and its resultant formula to help you find the length of the hypotenuse of a right triangle when you know the measurements of the triangle's two other sides.

A Right Triangle

Imagine that you are standing in front of a tree. You are standing 10 feet away from the tree. You are looking at the very top of the tree. You see a huge, delicious plum hanging from one of the branches. This plum happens to be hanging 30 feet in the air. Thirty feet is a bit too high for you to simply reach up and grab the plum. What will help you to get to this delectable piece of fruit? You need a ladder. But what size ladder do you need?

To figure this out, you can use math. If you drew out your situation, you would see that you, the tree, and the fruit form a right triangle. Your right triangle is a triangle with one 90 degree angle. This 90 degree angle is where the tree comes out from the ground. We can mark this angle by drawing a little square box at that angle like this.


You know the lengths of the two sides of the right triangle. The side that you need to know the length of is the distance between you and the top of the tree. This is where the ladder will go. This side is the hypotenuse of the right triangle, the side opposite the 90 degree angle. I've labeled it with the variable c to represent our as-of-yet-unknown hypotenuse. This is the variable that we need to solve for. We can solve for our hypotenuse by using the Pythagorean Theorem.

The Pythagorean Theorem

The Pythagorean Theorem tells us that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. In formula form, it is a^2 + b^2 = c^2, where a and b are the two sides of the right triangle and c is the hypotenuse. In our tree problem, our two sides are the 10 feet and the 30 feet; the hypotenuse is the c. When we use the Pythagorean Theorem to help us solve our problem, we plug in our values for a and b and then we solve the formula for c. Let's take a look.

Finding the Hypotenuse

It doesn't matter which side is a and which side is b. We can label the 10 feet as side a and the 30 feet as side b. As long as we don't make the mistake of labeling the hypotenuse as one of the sides, we're okay.

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