Geometry · Lesson 16 of 18

Pythagoras' Theorem

In a right-angled triangle, the square on the longest side equals the sum of the squares on the other two sides: a squared plus b squared equals c squared.

Secondary
Updated 14 November 2026

What you will learn

State Pythagoras’ theorem.
Find the hypotenuse and a shorter side.
Test whether a triangle has a right angle.

Before you start: you should know squares and square roots and what a right-angled triangle is. See Triangles.

The idea

In a right-angled triangle, the longest side is opposite the right angle. It is called the hypotenuse. Pythagoras’ theorem says that if you draw a square on each side, the two smaller squares have a total area equal to the biggest square.

Squares on the sides of a 3, 4, 5 triangle. The areas are 9, 16 and 25, and 9 + 16 = 25.

In symbols, with shorter sides $a$ and $b$ and hypotenuse $c$ :

$a^2 + b^2 = c^2.$

Finding the hypotenuse

Square the two shorter sides, add them, then take the square root.

A triangle with shorter sides 5 and 12. Work out 25 + 144 = 169, then the square root gives 13.

Finding a shorter side

Subtract instead: $a^2 = c^2 - b^2$ . The hypotenuse is always the longest side, so a shorter side must come out smaller than it.

The hypotenuse is 10 and one side is 8. The missing side is the square root of 100 minus 64, which is 6.

Testing for a right angle

If the sides of a triangle satisfy $a^2 + b^2 = c^2$ with $c$ the longest, it has a right angle. Whole-number sets like $3, 4, 5$ and $5, 12, 13$ are called Pythagorean triples.

Worked examples

Example 1: Hypotenuse

A right-angled triangle has shorter sides $9$ cm and $12$ cm. Find the hypotenuse.

$9^2 + 12^2 = 81 + 144 = 225$ .
$\sqrt{225} = 15$ .

Answer: $15$ cm

Example 2: Shorter side

A right-angled triangle has a hypotenuse of $13$ cm and one shorter side of $5$ cm. Find the other.

$13^2 - 5^2 = 169 - 25 = 144$ .
$\sqrt{144} = 12$ .

Answer: $12$ cm

Example 3: A diagonal

A screen is $5$ cm by $7$ cm. Find its diagonal to 1 decimal place.

$5^2 + 7^2 = 25 + 49 = 74$ .
$\sqrt{74} \approx 8.602$ .

Answer: about $8.6$ cm

Tip: always check that the hypotenuse is the longest side. If your answer for a shorter side is longer than the hypotenuse, you added when you should have subtracted.

Why it matters

Builders use the theorem to check corners are square, surveyors use it to find distances they cannot measure directly, and satnavs and games use it to work out the distance between two points. It works because the plane is flat, and its idea of distance extends to maps, screens and 3D space.

Practice

Your turn. Type or choose an answer, then press Check. Getting one wrong is fine, that is what practice is for. You can try as many times as you like.