Finding the Hypotenuse Using Pythagoras' Theorem
(KS3, Year 7)

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 = √(a2 + b2)
Don't forget: √ means square root and a2 = a × a (a squared), b2 = 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 = √(52 + 122) = √((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 = √(a2 + b2) to find the length of the hypotenuse, c. This is just a rearrangement of the more memorable formula, a2 + b2 = c2 (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.

Lesson Slides

The slider below gives a real example of how to find the hypotenuse using Pythagoras' theorem. Open the slider in a new tab

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 ,c2, will not always be a square number. When you square root c2, 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:

a2 + b2 = c2 12 + 22 = c2 1 + 4 = c2 5 = c2
Taking the square root gives c = 2.236, but it would be neater to write √5.
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See Also

What is a right triangle? What is a square root? What is a square number?