## The Lesson

The length of a shorter side of a right triangle (called a leg or cathetus) can be found from Pythagoras' theorem, if the length of the hypotenuse and another side is known. The length of the side is is found using the formula:In the formula,

**a**and

**b**are the lengths of the shorter sides and

**c**is the length of the hypotenuse. The image below shows what we mean:

## How to Find the Hypotenuse Using Pythagoras' Theorem

## Question

What is the unknown length, a, of the right triangle below?## Step-by-Step:

# 1

Start with the formula:

a = √(c

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

Find the lengths of the sides from the right triangle.
In our example, the known side lengths are b = 6 and h = 10.

# 3

Substitute b = 6 and c = 10 into the formula.

a = √(10

^{2}− 6^{2}) = √((10 × 10) − (6 × 6)) c = √(100 − 36) = √64 a = 8## Answer:

The length of the shorter side, a, is 8.## A Real Example of How to Find a Shorter Side Using Pythagoras' Theorem

In the example above, we have used the formula**a = √(c**to find the length of the side, a. 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 b and c with numbers) and then rearrange to find the side, a. There are two shorter sides, a and b. It does not really matter which of the shorter sides is labelled a and which is labelled b. The formula

**b = √(c**can also be used.

^{2}− a^{2})## Interactive Widget

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

Pythagoras' theorem states:Subtract

**b**from both sides:

^{2}Take square roots of both sides:

This gives the formula for finding the length of the unknown shorter side when the hypotenuse and the other shorter side are known:

## Top Tip

## Leaving the Answer as a Square Root

To find the side, a, you have to find the square root of a^{2}. Unless a

^{2}is a square number, a 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 b = 1 and c = 2:

a

Taking the square root gives a = 1.732, but it would be neater to write √3.
^{2}+ b^{2}= c^{2}a^{2}+ 1^{2}= 2^{2}a^{2}= 2^{2}− 1^{2}a^{2}= 4 − 1 a^{2}= 3