## The Lesson

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**to find the length of the hypotenuse, c. This is just a rearrangement of the more memorable formula,

^{2}+ b^{2})**a**(see

^{2}+ b^{2}= c^{2}**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.

## Interactive Widget

Here is an interactive widget to help you learn about Pythagoras' theorem.## Rearranging Pythagoras' Theorem

Pythagoras' theorem states:Taking square roots of both sides:

This gives the formula for finding the hypotenuse when the other sides are known:

## Top Tip

## Leaving the Answer as a Square Root

The square of the hypotenuse ,c^{2}, will not always be a square number. When you square root c

^{2}, to find c, the answer will not be a whole number. It is sometimes best to leave the answer as a square root (also called a surd). For example, find the hypotenuse for the right triangle below, where a = 1 and b = 2:

a

Taking the square root gives c = 2.236, but it would be neater to write √5.
^{2}+ b^{2}= c^{2}1^{2}+ 2^{2}= c^{2}1 + 4 = c^{2}5 = c^{2}