Pythagoras' Theorem — GCSE Maths Complete Guide

The Theorem and What It Actually Says

Pythagoras' theorem states that in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. As a formula: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse — the longest side, always found directly opposite the right angle.

It is one of the most reliable sources of marks on a GCSE paper. It appears on both tiers and every board, sometimes as a stand-alone 2 or 3 mark question, and often hidden inside a bigger one: the diagonal of a rectangle, the slant height of a cone, the distance between two coordinates, or the first step of a trigonometry problem. It sits in the same strand as SOHCAHTOA, and both are revised together on our Pythagoras and trigonometry topic page.

One restriction matters above everything: the theorem only works when the triangle has a right angle. If the diagram shows no right-angle mark and the question does not describe one (a wall meeting the ground, a ladder against a vertical wall), Pythagoras does not apply.

Finding the Hypotenuse

When the two shorter sides are known and the hypotenuse is the unknown, the routine is: square both sides, add, then square-root. Write each stage on its own line — on a 3-mark question there is typically a method mark for the squaring-and-adding step, so the working earns marks even if a later slip costs the final answer.

Always finish with a sanity check: the hypotenuse must be longer than either of the other two sides, but shorter than the two of them added together. An answer of 100 instead of 10 means you forgot the square root; an answer shorter than a given side means you subtracted instead of adding.

Finding a Shorter Side

When the hypotenuse is known and one of the shorter sides is the unknown, rearrange the formula: $a^2 = c^2 - b^2$. Square the hypotenuse, subtract the square of the known side, then square-root.

Most answers will not be whole numbers, and that is fine. If the sides were 5 cm and 7 cm, the hypotenuse would be $\sqrt{25 + 49} = \sqrt{74} \approx 8.60$ cm. Keep the exact value on your calculator until the final line, then round to the accuracy the question asks for — or leave it as a surd like $\sqrt{74}$ if the question says "exact".

The Big Mistake: Adding When You Should Subtract

The single most common Pythagoras error at GCSE is running the hypotenuse routine on a shorter-side question. A student sees two numbers, squares them, adds, square-roots — and gets a confident, wrong answer that the examiner cannot rescue with method marks, because the method itself was wrong.

The decision comes down to one question: is the unknown the hypotenuse or not? Finding the hypotenuse: add the squares. Finding either of the other sides: subtract the smaller square from the hypotenuse's square. Since the hypotenuse is opposite the right angle, you can settle this from the diagram before touching the numbers.

There is also a built-in alarm. If you find a "shorter side" by adding, your answer comes out longer than the hypotenuse — which is impossible, since the hypotenuse is the longest side. In the example above, adding would give $\sqrt{169 + 25} = \sqrt{194} \approx 13.9$ cm, longer than the 13 cm hypotenuse. Any time a shorter side comes out longer than the hypotenuse, go back and subtract.

Pythagorean Triples Worth Recognising

A Pythagorean triple is a set of three whole numbers that satisfies the theorem, like 3-4-5 (since $9 + 16 = 25$). Recognising the common ones saves real time: spot that a triangle has sides 5 and 12 with a right angle between them and you can write 13 without a calculator, which is exactly what non-calculator papers are testing.

Multiples of a triple are also triples. Doubling 3-4-5 gives 6-8-10 (the first worked example above), and 9-12-15, 30-40-50 and so on all work. Examiners lean on scaled 3-4-5 and 5-12-13 triangles constantly, so check for a scale factor when the numbers look familiar.

Distance Between Two Coordinates

One disguise Pythagoras wears on Higher papers (and increasingly on Foundation) is the distance between two points on a coordinate grid. The trick is to see the invisible right-angled triangle: the horizontal distance between the points is one side, the vertical distance is the other, and the straight line joining the points is the hypotenuse.

For points $(x_1, y_1)$ and $(x_2, y_2)$, the distance is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. You do not need to memorise this as a formula if you understand where it comes from — sketch the two points, draw the horizontal and vertical legs, and run ordinary Pythagoras.

Pythagoras in 3D (Higher Tier)

Higher papers extend the theorem into three dimensions, most often asking for the longest diagonal of a cuboid — the line from one corner to the opposite corner through the middle of the box. The method is to apply Pythagoras twice: once flat, once standing up.

First find the diagonal of the base using the length and width. That base diagonal then becomes one leg of a new right-angled triangle whose other leg is the height, and whose hypotenuse is the 3D diagonal you want. If you prefer one step, the two applications combine into $d^2 = a^2 + b^2 + c^2$ for a cuboid with edges $a$, $b$ and $c$ — but showing both stages makes your working easier to follow and easier to award marks to.

The same two-step idea handles pyramids and cones: find a horizontal distance across the base first, then use it with the vertical height.

Exam Technique and What to Revise Next

Pythagoras questions are marked generously if the working is visible: state the formula or the substitution, show the squaring, show the addition or subtraction, then the root. Round only at the end, and leave surds in when the question asks for an exact answer — $\sqrt{74}$ is exact, 8.6 is not.

Know when Pythagoras is the tool and when it is not. Two known sides, no angle involved: Pythagoras. An angle (other than the right angle) enters the question, or you are asked to find one: that is trigonometry, covered in our SOHCAHTOA guide. No right angle anywhere: neither applies directly, and Higher students reach for the cosine rule instead.

Exam Ladder generates Pythagoras questions from simple hypotenuse-finding through to 3D diagonals, each with a fully worked solution, drawn from a bank of over 280,000 generated practice questions across 70 GCSE topics. Practise the add-or-subtract decision until it is instant — it is the whole question.