Pythagoras' Theorem: GCSE Maths Formula, Proof & Examples

Pythagoras’ theorem is one of the most important results in GCSE Maths. This guide is part of our complete GCSE Maths revision guide — start there for an overview of all topics and revision strategies. The theorem connects the three sides of a right-angled triangle through an elegant relationship that’s been around for thousands of years. Finding a missing side? Calculating a diagonal in 3D? You’ll need this.

The Formula

In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:

a² + b² = c²

Here, c is the hypotenuse (the longest side, always opposite the right angle) and a and b are the two shorter sides. Get the hypotenuse wrong and everything falls apart. Seriously. The pattern in mock-paper analyses is consistent: students who misidentify the hypotenuse lose every mark on the question, because the substitution is wrong from the very first line.

How to Identify the Hypotenuse

The hypotenuse is always:

• The longest side of the triangle
• The side opposite the right angle
• The side that is not touching the right angle

Label the right angle first. The side across from it is always c. The other two sides can be a and b in either order — it genuinely doesn’t matter which is which.

Worked Examples

Example 1: Finding the Hypotenuse

A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the hypotenuse.

Known values: a = 6, b = 8

Find: c

a² + b² = c²

36 + 64 = c²

100 = c²

c = 10 cm

Example 2: Finding a Shorter Side

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

Known values: c = 13, a = 5

Find: b

a² + b² = c²

25 + b² = 169

b² = 144

b = 12 cm

Notice how you rearrange when finding a shorter side. Instead of adding, you subtract from the hypotenuse squared.

Example 3: A Non-Integer Answer

A right-angled triangle has sides of 4 cm and 7 cm. Find the hypotenuse.

a² + b² = c²

16 + 49 = c²

65 = c²

c = 8.06 cm (to 3 significant figures)

Not every answer will be a whole number. When it isn’t, give your answer to an appropriate degree of accuracy. The question will usually specify decimal places or significant figures.

Pythagorean Triples

A Pythagorean triple is a set of three whole numbers that satisfy Pythagoras’ theorem. The ones you’ll see most at GCSE:

• 3, 4, 5 (and multiples like 6, 8, 10 and 9, 12, 15)
• 5, 12, 13
• 8, 15, 17
• 7, 24, 25

Recognising these saves time. Spot that two given sides belong to a known triple? Write down the answer immediately. No calculation needed.

3D Pythagoras (Higher Tier)

At Higher tier, you may need to find the length of a diagonal inside a cuboid or other 3D shape. The method uses Pythagoras’ theorem twice.

A common stumbling block: students try to do it in one step and get confused. Break it down — find a face diagonal first, then use that with the remaining dimension.

Example: Diagonal of a Cuboid

A cuboid has dimensions 3 cm by 4 cm by 12 cm. Find the length of the space diagonal (the diagonal running from one corner to the opposite corner through the interior).

Step 1: Find the diagonal of the base using length and width.

d² = 3² + 4² = 9 + 16 = 25, so d = 5 cm

Step 2: Use this base diagonal and the height to find the space diagonal.

D² = 5² + 12² = 25 + 144 = 169, so D = 13 cm

You can also combine both steps into a single formula: D² = a² + b² + c², where a, b, and c are the three dimensions. Here, D² = 9 + 16 + 144 = 169, giving D = 13 cm.

Distance Between Two Coordinates

Pythagoras’ theorem also finds the straight-line distance between two points on a coordinate grid. Given two points (x₁, y₁) and (x₂, y₂):

distance = square root of [(x₂ - x₁)² + (y₂ - y₁)²]

This is just Pythagoras applied to the horizontal and vertical differences between the points — they form the two shorter sides of a right-angled triangle.

Example

Find the distance between (1, 3) and (7, 11).

Horizontal difference: 7 - 1 = 6

Vertical difference: 11 - 3 = 8

distance² = 6² + 8² = 36 + 64 = 100

distance = 10 units

Common Mistakes to Avoid

A mistake that surfaces every single exam season: writing down the formula correctly, substituting correctly, calculating a² + b² correctly… then writing that number as the final answer. The square root is forgotten. If a “length” comes out as 169 cm for a triangle with sides of 5 and 12, something’s gone wrong. Always ask: does this answer make sense as an actual length?

Not Identifying the Hypotenuse

The most frequent error is treating one of the shorter sides as the hypotenuse. Always check: which side is opposite the right angle? That’s c. Put a shorter side as c and your answer will be wrong.

Squaring Errors

When working without a calculator, take care with your arithmetic. Common slips include calculating 7² as 21 instead of 49, or 12² as 124 instead of 144. Write each squaring step out separately.

Adding When You Should Subtract

When finding a shorter side, you need to rearrange to b² = c² - a². Students sometimes add the two known squares instead of subtracting. Remember: add when finding the hypotenuse. Subtract when finding a shorter side.

Applying Pythagoras to Non-Right-Angled Triangles

The theorem only works for right-angled triangles. No right angle stated or shown? Check whether you should be using the cosine rule or sine rule instead. Sometimes you need to split a shape into right-angled triangles first.

Exam Technique

Draw and label the triangle. Even if the question provides a diagram, redrawing it with your own labels helps you identify the hypotenuse and avoid errors.

Show every step. Write the formula, substitute your values, show the squaring, and take the square root. Examiners give method marks for each step.

State your units. Always include units in your final answer. Sides in centimetres? Answer in centimetres.

A useful five-step checklist when a Pythagoras question feels stuck: formula, numbers in, calculate, square root, units. Five steps, every time.

Summary

Pythagoras’ theorem states that in a right-angled triangle, a² + b² = c² where c is the hypotenuse. Learn to identify the hypotenuse, practise rearranging for shorter sides, and memorise Pythagorean triples for quick answers. At Higher tier, extend the method into three dimensions and coordinate geometry.

Related GCSE Maths Guides

• Geometry and measures guide — angles, shapes, transformations, and trigonometry
• Cosine rule guide — solving non-right-angled triangles with SAS and SSS
• Circle theorems guide — all 8 circle theorems with worked examples

How to Use This Guide

Read through the worked examples first, then cover the solutions and try them yourself. If you got them right without looking, move on to 3D Pythagoras and coordinate distance — that’s where the Higher tier marks are. If you’re still shaky on the basics, drill Example 1 and Example 2 types until they’re automatic. The formula should become muscle memory. Then try GCSE Maths questions on UpGrades to find and fix any remaining weak spots.

If you want structured practice on Pythagoras’ theorem and other GCSE Maths topics, UpGrades generates AI-powered practice questions that adapt to your level, so you spend your revision time where it counts. Try it free today.