# How to Find the Hypotenuse Using Pythagoras' Theorem

## Finding the Hypotenuse Using Pythagoras' Theorem

The length of the longest side of a right triangle (called the hypotenuse) can be found from Pythagoras' theorem, if the lengths of the other two sides are known.

The hypotenuse is found using the formula:

In the formula, **c** is the length of the hypotenuse and **a** and **b** are the lengths of the other sides. The image below shows what we mean:

## How to Find the Hypotenuse Using Pythagoras' Theorem

### Question

What is the hypotenuse of the right triangle below?

### Step-by-Step:

# 1

Start with the formula:

c = √(a^{2} + b^{2})

**Don't forget:** √ means square root **and** a^{2} = a × a (a squared), b^{2} = b × b.

# 2

Find the lengths of the sides from the right triangle.

In our example, the two shorter side lengths are a = 5 and b = 12.

# 3

Substitute a = 5 and b = 12 into the formula.

c = √(5^{2} + 12^{2}) = √((5 × 5) + (12 × 12))

c = √(25 + 144) = √169

c = 13

### Answer:

The length of the hypotenuse is 13.

## A Real Example of How to Find the Hypotenuse Using Pythagoras' Theorem

In the example above, we have used the formula **c = √(a ^{2} + b^{2})** to find the length of the hypotenuse, c.

This is just a rearrangement of the more memorable formula, **a ^{2} + b^{2} = c^{2}** (see

**Note**).

If you find this simpler formula easier to remember, use it!

Substitute in the lengths you know (replace the letters a and b with numbers) and then rearrange to find the hypotenuse, c.