Pythagoras' Theorem
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
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In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
- a squared plus b squared equals c squared
- The hypotenuse is the longest side
- 3, 4, 5 is a Pythagorean triple
🧠 Quick quiz
Three questions. Score 2 or more to complete the topic.
What is Pythagoras' Theorem?
Within Mathematics, which covers numbers, algebra, geometry, trigonometry and statistics from first principles to exam level, this topic explains Pythagoras' Theorem. Maths builds step by step, so each topic here gives you a rule, a picture and a quick check you can use on homework and in exams.
Check that you can explain Pythagoras' Theorem in your own words and give one example from everyday life. If you can do that, you have understood the core idea.
Understanding Pythagoras' Theorem gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
Pythagoras' Theorem in detail
In a right-angled triangle, the square of the hypotenuse, the longest side opposite the right angle, equals the sum of the squares of the other two sides: a² + b² = c². A 3-4-5 triangle works because 9 + 16 = 25.
To find the hypotenuse, add the squares and take the square root. To find a shorter side, subtract. The converse also holds, so if the side lengths satisfy the equation, the triangle is right-angled. The theorem is used in construction, navigation, screen sizes and finding distances in coordinate geometry and 3D shapes.
How the interactive helps you understand Pythagoras' Theorem
Watch the right-angled triangle in the interactive. Use any sliders or buttons it offers to change the values and see how the right-angled triangle responds.
Before you change anything, predict the result. Then change one value at a time and check whether your prediction was right. This is a quick way to build a correct mental model of Pythagoras' Theorem.
Real-life uses and examples
- Spot Pythagoras' Theorem in exam questions, where it often sits inside a short word problem.
- Check a homework answer about Pythagoras' Theorem by estimating first, then working it out step by step.
- Use Pythagoras' Theorem when comparing prices, discounts or recipe quantities in everyday life.
Common mistakes to avoid
- Forgetting units, signs or brackets when you apply Pythagoras' Theorem.
- Memorising the steps for Pythagoras' Theorem without checking why they work.
- Learning the definition without testing it. Use the quiz and the interactive to check that you can apply the idea.
Frequently asked questions about Pythagoras' Theorem
What is Pythagoras' Theorem?
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
What are the most important points about Pythagoras' Theorem?
The main points are: a squared plus b squared equals c squared; The hypotenuse is the longest side; 3, 4, 5 is a Pythagorean triple.
Why is Pythagoras' Theorem useful?
Understanding Pythagoras' Theorem gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
How can I practise Pythagoras' Theorem?
Open the interactive at the top of this page and change the controls one at a time. Predict the result before each change, then take the three-question quiz. A score of two or more marks the topic as complete on this device.
Which topics are related to Pythagoras' Theorem?
Related topics in Mathematics include Sine and Cosine Rules, Bearings, Surface Area.