Pythagoras’s theorem

A theorem specific to right-angled triangles

Pythagoras’ theorem applies only to right-angled triangles, that is, to triangles with a 90-degree angle.

The Hypotenuse

In a right-angled triangle, the longest side, opposite the right angle, is called the hypotenuse.

The statement

If a triangle ABC is right-angled at A, then:

BC² = AB² + AC²

(the square of the hypotenuse is equal to the sum of the squares of the other two sides, known as the legs).

Practical example

A right-angled triangle has sides opposite the right angle measuring 3 cm and 4 cm. We are looking for the hypotenuse:

BC² = 3² + 4² = 9 + 16 = 25 BC = √25 = 5 cm

This is the famous ‘3-4-5’ triangle, which is widely used because its measurements are whole numbers.

Finding a side of the right-angled triangle

If we know the hypotenuse and one side of the right-angled triangle, we subtract rather than add. Example: hypotenuse = 13 cm, one side = 5 cm.

The other side² = 13² - 5² = 169 - 25 = 144, so the other side = √144 = 12 cm.

Common pitfall

Frequent mistake: adding the squares even when the triangle is NOT a right-angled triangle (the theorem does not apply in this case). Another common mistake: getting the sides mixed up and subtracting instead of adding, or vice versa. Always identify the hypotenuse first (the longest side, opposite the right angle) before calculating.