Pythagoras' Theorem — An Examiner's Guide

Pythagoras' Theorem is one of those topics that turns up on virtually every GCSE Maths Higher paper and most Foundation papers — usually for three or four marks at minimum, sometimes embedded inside a longer question for six or more. It's also one of the easiest to lose marks on, because the question almost never says "use Pythagoras". You have to spot when to use it. This guide walks through the theorem itself, the four flavours of question you'll see, and the mistakes I see most often when marking exam scripts.

The theorem itself

For any right-angled triangle with shorter sides a and b and hypotenuse c:

a² + b² = c²

The hypotenuse is the longest side, and it's always opposite the right angle. If you remember nothing else, remember that — the formula only works if c is the hypotenuse. Putting the wrong side in the wrong slot is the single most common mistake I see.

The two question types you'll meet

Every GCSE Pythagoras question — at any difficulty — boils down to one of two situations: you're either finding the hypotenuse (the long side) or finding one of the shorter sides. The arithmetic is slightly different for each.

Finding the hypotenuse — when both shorter sides are given

Finding a shorter side — when the hypotenuse is given

This is where students slip up. If the hypotenuse is given and you need a shorter side, you have to rearrange the formula first:

a² = c² − b²

How to spot a Pythagoras question

Examiners rarely write "use Pythagoras". They give you a worded question and expect you to recognise when it applies. The triggers are:

• A right angle is mentioned, drawn, or implied — flagpoles, ladders against walls, ramps, rectangular fields, square rooms
• You're given two sides of a triangle and asked for the third
• The question asks for a distance between two points described by a horizontal and vertical movement
• A diagonal of a rectangle, square, or rectangular box is involved

Distance between two points

If two points have coordinates (x₁, y₁) and (x₂, y₂), the distance between them is found by drawing a right-angled triangle and using Pythagoras on the horizontal and vertical differences:

3D Pythagoras (Higher tier)

On Higher papers you'll sometimes see a 3D problem — usually finding the longest diagonal inside a rectangular box (a cuboid). The trick: apply Pythagoras twice. First on the base of the box to get the diagonal across the floor; then again, treating that floor diagonal and the box's height as the two shorter sides of a new right-angled triangle.

For a deeper dive on 3D problems including ones involving angles, see my separate guide on 3D Trigonometry.

The mistakes that cost the most marks

Useful Pythagorean triples to recognise

If a question gives you two sides and they're part of one of the classic Pythagorean triples, you can save calculator time by spotting the third side instantly. The four worth committing to memory:

• 3, 4, 5 — and any multiple (6, 8, 10; 9, 12, 15; 30, 40, 50)
• 5, 12, 13
• 8, 15, 17
• 7, 24, 25

If you spot one of these in a question, you can write down the third side without working — though I'd still recommend writing the Pythagoras step out for the method mark.