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. It connects the three sides of a right-angled triangle through an elegant relationship that has been used for thousands of years. Whether you are finding a missing side in a simple triangle or calculating the length of a diagonal in three dimensions, this theorem is essential.

The Formula

In any right-angled triangle, the square of the hypotenuse is equal to 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. Identifying the hypotenuse correctly is the single most important step when using this theorem.

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

If you 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 since it does not 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 the formula 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 is not, give your answer to an appropriate degree of accuracy. The question will usually tell you whether to round to a number of decimal places or significant figures.

Pythagorean Triples

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

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

Recognising these triples saves time in exams. If you spot that two given sides belong to a known triple, you can write down the answer immediately without any calculation.

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.

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 that runs from one corner to the opposite corner through the interior).

Step 1: Find the diagonal of the base using the 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 lets you find 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 simply Pythagoras’ theorem applied to the horizontal and vertical differences between the two points, which 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

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 is c. If you put a shorter side as c, your answer will be wrong.

Forgetting to Square Root at the End

Students often calculate a² + b² correctly but then leave the answer as c² rather than taking the square root. Your final answer must be c, not c². Always check whether your answer seems reasonable for a length.

Squaring Errors

When working without a calculator on non-calculator papers, 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: you only add when finding the hypotenuse. When finding a shorter side, subtract.

Applying Pythagoras to Non-Right-Angled Triangles

The theorem only works for right-angled triangles. If the question does not state or show a right angle, 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 the units in your final answer. If the sides are given in centimetres, your answer is in centimetres too.

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 remember 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

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