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 = √(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.

Slider

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

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See Also

What is a right triangle? What is a square root? What is a square number? Learn more about Pythagoras' theorem ( interactive widget)