Finding a Shorter Side Using Pythagoras' Theorem
(KS3, Year 7)
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:

pythagoras formula shorter side 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:pythagoras_triangle

How to Find the Hypotenuse Using Pythagoras' Theorem

Question

What is the unknown length, a, of the right triangle below?
pythagoras shorter side question

Step-by-Step:

1

Start with the formula:
a = √(c2 − b2)
Don't forget: √ means square root and c2 = c × c (c squared), b2 = 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 = √(102 − 62) = √((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 = √(c2 − b2) to find the length of the side, a. 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 b and c with numbers) and then rearrange to find the side, a.

Lesson Slides

The slider below gives a real example of how to find a shorter side using Pythagoras' theorem.
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 = √(c2 − a2) can also be used.

Interactive Widget

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

Rearranging Pythagoras' Theorem

Pythagoras' theorem states:

pythagoras formula mini Subtract b2 from both sides:

pythagoras formula a squared Take square roots of both sides:

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

pythagoras formula shorter side mini

Top Tip

Leaving the Answer as a Square Root

To find the side, a, you have to find the square root of a2. Unless a2 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:

a2 + b2 = c2

a2 + 12 = 22

a2 = 22 − 12

a2 = 4 − 1

a2 = 3

Taking the square root gives a = 1.732, but it would be neater to write √3.
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This page was written by Stephen Clarke.