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Here’s How To Find the Hypotenuse of a Right Triangle

The Pythagorean Theorem is an important theorem, as it comes up often in high school math. It makes the question of how to find the hypotenuse of a right triangle is easy to answer.

Plus, we can use it to find the base of a special right triangle and apply the converse of the Pythagorean Theorem. Let’s review this basic but foundational concept in trigonometry and how it relates to finding the hypotenuse of a right triangle.

Defining the Hypotenuse

First, let's demystify this term. "Hypotenuse" is simply a term that means "the longest side of a right triangle." The hypotenuse is the opposite side of the right angle in the triangle. It’s also the longest side of the triangle.

If a problem asks you to calculate the length of hypotenuse c in a triangle with side a, side b, and hypotenuse c, then you are working with a right-angled triangle.

How the Pythagorean Theorem Applies

The Pythagorean Theorem is a well-known theorem developed by a Greek mathematician named Pythagoras around 500 BC. It related the length of the hypotenuse of a right triangle to the lengths of the legs (the legs of the triangle are the two shorter sides). The Pythagorean Theorem states that for any right triangle, the sum of the squares of the lengths of the legs is always equal to the square of the length of the hypotenuse.

The Pythagorean Theorem states that:

how to find hypotenuse: The Pythagorean Theorem

Because of the Pythagorean Theorem, it is easy to find the hypotenuse of a right triangle if we are given the sides of a right triangle. In fact, if we know the lengths for any two sides (leg A, leg B, and hypotenuse C) we can easily figure out the missing side by applying the formula.

This can help you find any missing side. But for now, let's see an example where we know the length of the sides and want to find the hypotenuse:

how to find hypotenuse: Diagram of a triangle with an unknown hypotenuse

Here we know the length of the side (a = 3 and b = 4). If we substitute it into the formula, we get:

how to find hypotenuse: The Pythagorean Theorem

how to find hypotenuse: Formula for finding the hypotenuse

how to find hypotenuse: Formula for finding the hypotenuse

Formula for finding the hypotenuse

And by taking the square root on both sides, we figure out that the hypotenuse c = 5!

Formula for finding the hypotenuse

Equation showing the value of the hypotenuse

Summary for Finding the Hypotenuse

The Pythagorean Theorem helps us calculate the hypotenuse of a right triangle if we know the sides of the triangle.

There are more advanced trigonometric functions that allow us to calculate the third side of a triangle, even non-right triangles, given a particular degree angle and side length.

But either way, practice applying the Pythagorean Theorem until you feel confident with right triangles. Then, you can learn how to find the third side of any triangle.

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