Pythagoras' Theorem in 2D and 3D — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers Pythagoras' theorem: the relationship between the three sides of a right-angled triangle. By the end of this guide you should be able to find the hypotenuse, find a shorter side, decide whether a triangle is right-angled, and apply the theorem to problems set in real contexts.

You should also be able to use it on a coordinate grid to find the distance between two points, recognise the common Pythagorean triples, and extend the method into three dimensions to find the space diagonal of a cuboid.

The organising idea is a single decision that controls everything: is the side you are looking for the hypotenuse, or a shorter side? If it is the hypotenuse, you square the two known sides and add. If it is a shorter side, you square both and subtract. Every question in this topic is that choice, followed by a square root. Getting it wrong is the single largest source of lost marks, and it is entirely avoidable by identifying the hypotenuse before writing anything down.

Key terms and definitions

Right-angled triangle — a triangle containing an angle of exactly 90°. Pythagoras' theorem applies to these and to no others.

Hypotenuse — the side opposite the right angle. It is always the longest side of the triangle.

Shorter sides (or legs) — the two sides that form the right angle.

Pythagoras' theorem — a² + b² = c², where c is the hypotenuse and a and b are the shorter sides.

Pythagorean triple — a set of three whole numbers that fit the theorem exactly, such as 3, 4, 5.

Space diagonal — in a cuboid, the line from one corner to the opposite corner, passing through the middle of the solid.

Core concepts

What the theorem actually says

The theorem is a statement about areas. Draw a square on each side of a right-angled triangle, and the area of the square on the hypotenuse equals the areas of the other two squares added together. That is why the relationship involves squares rather than the lengths themselves.

Written as a formula, a² + b² = c², with c always the hypotenuse.

It works only in right-angled triangles. In a triangle with no right angle, the relationship fails, and a different tool — the cosine rule — is needed instead.

Identifying the hypotenuse

The hypotenuse is the side opposite the right angle, and it is always the longest side. In a worded question it is often the slanted or diagonal length: the ladder rather than the wall, the distance between two points rather than the horizontal and vertical steps.

Identify it before doing any arithmetic. Everything else in the question depends on it.

Finding the hypotenuse: square and add

When the two shorter sides are known, square them, add, and take the square root.

For shorter sides of 6 cm and 8 cm: 6² + 8² = 36 + 64 = 100, so the hypotenuse is √100 = 10 cm.

Check the answer for sense. The hypotenuse must be longer than either shorter side but shorter than their sum — here, between 8 and 14. An answer outside that range means an error has been made.

Finding a shorter side: square and subtract

When the hypotenuse and one shorter side are known, square both, subtract the smaller square from the larger, and take the square root.

For a hypotenuse of 13 cm and a shorter side of 5 cm: 13² − 5² = 169 − 25 = 144, so the other shorter side is √144 = 12 cm.

The order of the subtraction matters. Taking 169 from 25 gives a negative number, and a negative result is the clearest possible signal that the hypotenuse was not correctly identified. Always subtract from the square of the longest side.

Here the answer must be less than the hypotenuse, which is the sense check that mirrors the one above.

Pythagorean triples

A few sets of whole numbers satisfy the theorem exactly, and recognising them saves time: 3, 4, 5; 5, 12, 13; 8, 15, 17; and 7, 24, 25.

Multiples work too, so 6, 8, 10 and 9, 12, 15 are both versions of the 3, 4, 5 triple.

Most exam answers are not whole numbers, and there is nothing wrong with an answer such as 8.6 cm. Do not force a value to fit a triple.

Testing whether a triangle is right-angled

The theorem works in reverse. Square the longest side, square the other two and add them, then compare.

For 9, 12, 15: 9² + 12² = 81 + 144 = 225, and 15² = 225. They match, so the triangle is right-angled.

For 5, 6, 8: 25 + 36 = 61, but 8² = 64. They do not match, so the triangle has no right angle.

Distance between two points

On a coordinate grid, the horizontal and vertical gaps between two points form the shorter sides of a right-angled triangle, and the distance between the points is the hypotenuse.

From (1, 2) to (4, 6): the horizontal gap is 4 − 1 = 3 and the vertical gap is 6 − 2 = 4. The distance is √(3² + 4²) = √25 = 5.

Take the difference in each coordinate, not the coordinates themselves. Using 4 and 6 rather than 3 and 4 is a frequent slip.

Three dimensions

Finding the space diagonal of a cuboid means applying the theorem twice.

First find the diagonal across the base, using the length and width as the shorter sides. Then use that base diagonal together with the height as the two shorter sides of a second, upright right-angled triangle, whose hypotenuse is the space diagonal.

The first answer is an intermediate value, so keep it unrounded — or better, keep it squared, since the second calculation needs its square anyway. Rounding at the halfway point is the commonest cause of a slightly wrong final answer.

Worked examples

Example 1: Finding a shorter side

A ladder 5 m long leans against a wall with its base 3 m from the wall. How far up the wall does it reach?

The ladder is the slanted side, so it is the hypotenuse. The height up the wall is a shorter side, which means squaring and subtracting.

5² − 3² = 25 − 9 = 16, so the height is √16 = 4 m.

The answer is less than the ladder's length, as it must be. ✓

Example 2: Testing for a right angle

A triangle has sides 8 cm, 15 cm and 17 cm. Is it right-angled?

The longest side is 17 cm, so compare 17² with the sum of the other two squares.

8² + 15² = 64 + 225 = 289, and 17² = 289.

They are equal, so the triangle is right-angled, and 8, 15, 17 is a Pythagorean triple.

Example 3: A space diagonal

A cuboid measures 3 cm by 4 cm by 12 cm. Find the length of its space diagonal.

First, the diagonal across the 3 cm by 4 cm base: 3² + 4² = 9 + 16 = 25, so that diagonal squared is 25.

Now use the base diagonal and the height of 12 cm as the two shorter sides of the upright triangle. Its hypotenuse squared is 25 + 12² = 25 + 144 = 169.

The space diagonal is √169 = 13 cm.

Working with the squared value throughout avoids rounding the intermediate answer of 5 cm, and here it also makes the arithmetic exact.

Common mistakes and how to avoid them

Adding when you should subtract. Decide first whether the unknown is the hypotenuse. Adding gives a longer side; subtracting gives a shorter one.

Subtracting the wrong way round. Always take the smaller square from the larger. A negative result means the hypotenuse was misidentified.

Forgetting the square root. The theorem gives the square of the unknown side. The length is the root of that.

Using the theorem on a triangle with no right angle. Check for the right angle first; without one, the cosine rule is the correct tool.

Using coordinates instead of their differences. The shorter sides are the horizontal and vertical gaps.

Rounding the intermediate answer in a 3D problem. Keep the base diagonal unrounded, or keep it squared.

Dropping the units. Lengths need cm, m or km, and the final accuracy mark often depends on them.

Exam technique for "Pythagoras' Theorem"

Mark the hypotenuse before you start. One word written on the page prevents the error that costs most marks in this topic.

Write the theorem out with the numbers substituted into it, such as 13² − 5² = 144, as a separate line. Method marks are awarded for a correct substitution even when the arithmetic that follows goes wrong.

Sense-check every answer: a hypotenuse must be longer than either shorter side, and a shorter side must be less than the hypotenuse.

Follow the rounding instruction exactly — "to 1 decimal place" or "to 3 significant figures" — and round only at the very end.

In three dimensions, say which triangle you are working with at each stage. Labelling the base diagonal makes a two-stage solution easy for the examiner to follow, and easy for you to check.

Quick revision summary

Pythagoras' theorem is a² + b² = c², with c the hypotenuse — the side opposite the right angle, and always the longest.

The whole topic turns on one decision. To find the hypotenuse, square the two shorter sides and add. To find a shorter side, square the hypotenuse and the known side and subtract, always taking the smaller square from the larger. Then take the square root.

Sense-check: the hypotenuse lies between the longer shorter side and the sum of both; a shorter side is always less than the hypotenuse.

Recognise the triples 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25, and their multiples — but do not force an answer to fit one.

To test for a right angle, compare the square of the longest side with the sum of the other two squares.

For the distance between two points, use the horizontal and vertical differences as the shorter sides.

For a space diagonal, apply the theorem twice — first across the base, then with the height — and keep the intermediate value unrounded.