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Pythagoras Theorem GCSE Maths

Pythagoras theorem is used to find missing side lengths in right-angled triangles. It is one of the most important GCSE geometry topics because it appears in many exam questions, including worded problems and questions involving shapes.

Video explanation

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What is Pythagoras theorem?

In a right-angled triangle, the two shorter sides and the longest side are connected by this rule:

a² + b² = c²

The longest side is called the hypotenuse. It is always opposite the right angle.

Exam tip: Before using Pythagoras, check that the triangle has a right angle. Pythagoras only works in right-angled triangles.

Finding the hypotenuse

If you know the two shorter sides, add their squares and then square root the answer.

Example 1: Find the hypotenuse

A right-angled triangle has shorter sides 3 cm and 4 cm.

c² = 3² + 4²
c² = 9 + 16
c² = 25
c = 5

The hypotenuse is 5 cm.

Example 2: Find the hypotenuse

A right-angled triangle has shorter sides 5 cm and 12 cm.

c² = 5² + 12²
c² = 25 + 144
c² = 169
c = 13

The hypotenuse is 13 cm.

Finding a shorter side

If you know the hypotenuse and one shorter side, subtract the squares and then square root the answer.

Example 3: Find a shorter side

A right-angled triangle has hypotenuse 10 cm and one shorter side 6 cm.

x² = 10² - 6²
x² = 100 - 36
x² = 64
x = 8

The missing shorter side is 8 cm.

Common mistake: Do not always add. Add when finding the hypotenuse. Subtract when finding a shorter side.

How to decide whether to add or subtract

  • If the missing side is the hypotenuse, add the squares.
  • If the missing side is a shorter side, subtract the squares.
  • The hypotenuse is always opposite the right angle.

Exam-style worded example

Example 4: Ladder against a wall

A ladder is 13 m long. The bottom of the ladder is 5 m from a wall. How high up the wall does the ladder reach?

The ladder is the hypotenuse because it is the longest side.

h² = 13² - 5²
h² = 169 - 25
h² = 144
h = 12

The ladder reaches 12 m up the wall.

Common mistakes in Pythagoras questions

  • Using Pythagoras when there is no right angle.
  • Choosing the wrong side as the hypotenuse.
  • Adding when the question needs subtraction.
  • Forgetting to square root at the end.
  • Rounding too early in multi-step questions.

Practice questions

  1. Find the hypotenuse when the shorter sides are 6 cm and 8 cm.
  2. Find the hypotenuse when the shorter sides are 9 cm and 12 cm.
  3. Find the shorter side when the hypotenuse is 13 cm and the other side is 5 cm.
  4. Find the shorter side when the hypotenuse is 17 cm and the other side is 8 cm.
  5. A ladder is 10 m long and the bottom is 6 m from a wall. How high does it reach?

Answers

  1. 10 cm
  2. 15 cm
  3. 12 cm
  4. 15 cm
  5. 8 m

Pythagoras FAQ

What is Pythagoras theorem?

Pythagoras theorem says that in a right-angled triangle, a² + b² = c², where c is the hypotenuse.

When can I use Pythagoras theorem?

You can use Pythagoras theorem only in right-angled triangles.

What is the hypotenuse?

The hypotenuse is the longest side of a right-angled triangle. It is always opposite the right angle.

Is Pythagoras on GCSE maths?

Yes. Pythagoras theorem is a common GCSE maths topic and appears on both Foundation and Higher papers.

Related GCSE geometry topics

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