Pythagoras’ theorem
Maths · Key Stage 3 · The School
The statement
In a RIGHT-ANGLED triangle, the square on the longest side equals the sum of the squares on the other two: a² + b² = c², where c is the HYPOTENUSE — always the side opposite the right angle, and always the longest. Identifying the hypotenuse before substituting is what prevents most errors with this.
Finding a missing side
To find the hypotenuse, square the two shorter sides, add them, and take the square root. To find a SHORTER side, square the hypotenuse, SUBTRACT the square of the known short side, then take the root. The two cases differ by one operation and mixing them up produces an answer longer than the hypotenuse, which is impossible and therefore self-checking.
When it does not apply
The theorem holds only for right-angled triangles. A triangle with no right angle needs different rules entirely. It is also worth knowing the converse: if a² + b² = c² for a triangle's sides, then it MUST be right-angled — which is how builders check a corner is square using a 3-4-5 triangle, without a protractor.
FINDING THE HYPOTENUSE: sides 6 cm and 8 cm. 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm. FINDING A SHORTER SIDE: hypotenuse 13 cm, one side 5 cm. Here you SUBTRACT: 13² − 5² = 169 − 25 = 144, so the other side is √144 = 12 cm. Adding instead would give √194 ≈ 13.9 cm — longer than the hypotenuse, which is impossible. That sense-check catches the commonest error in the topic.