Maths · KS3 and GCSE · Interactive lesson
Pythagoras’ theorem: finding a missing side
Connect square areas to side lengths in a right-angled triangle. Find the longest side or a shorter side and practise each method.
Choose your view
Explore an interactive example, then try the questions at your own pace.
Later KS3 / GCSE
By the end of this lesson: Use the square areas on a right-angled triangle to connect its side lengths.
The interactive diagram and controls are in Stage 2, “Explore one example”. Read the idea here, then try it next.
- A right angle is 90°. The longest side is across from it.
- Build a square on each side. The two smaller square areas add to the largest square area.
- a² means a × a. We are comparing areas before calculating a missing length.
Useful starting points
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What do the words mean?
- Hypotenuse
- The longest side, opposite the right angle. It is called c here.
- Squared (²)
- A number multiplied by itself: 3² = 3 × 3 = 9.
- Square root (√)
- √25 = 5 because 5 × 5 = 25. For a side length, use the positive square root.
Change the example
This is the lesson example. Your practice question uses its own numbers.
Something to try
Keep one shorter side the same. Increase the other. What happens to the longest side?
Notice what changes
How particle motion works +
Starts paused. Press Play, then add energy to see the warmer particles move faster.
Particles keep the same size. Liquid particles move around; solid particles vibrate around fixed positions. This is a simplified model, with the change in motion exaggerated to make it easier to see.
About the playback +
About this model
The squares are drawn to scale. Their areas are in cm²; side lengths are in cm. c is across from the right angle, which is marked by a little square. ≈ means the side length is rounded.
More about this diagram
Still picture. Play particle motion in Stage 2.
Square areas: 9 + 16 = 25 cm²
Still picture of the example from Stage 2. Use the back arrow to change it.
Work through it
Starting example: read the steps below
- Find the 90° corner. The longest side is c.
- The square on side a has area 3 × 3 = 9 cm².
- The square on side b has area 16 cm².
- Together: 9 + 16 = 25 cm², the area of the square on side c.
Try on your own
Your turn. No rush.
Question 1
Which side lies opposite the 90° angle?
About these questions
Questions appear in a mixed order. Work at your own pace and stop whenever you like.
These questions use their own numbers, separate from the lesson example. You can stop whenever you like. Answers last for this visit to the page.
A good place to pause
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Can you identify the longest side and choose whether to add or subtract the square areas?
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Notes for parents and tutors
These lessons cover selected KS3 and GCSE ideas for England; they are not a complete course. The level notes explain where GCSE methods or optional extensions begin. Maurice can help you choose material for your course and exam board.
The content draws on the DfE Maths curriculum, DfE Science curriculum and AQA GCSE Physics specification. Level labels are guidance, not exam-board endorsements. For AQA Radioactivity, calculating net decline as a ratio is Higher-tier work. Net means the overall change; ask Maurice whether this extension is relevant to your course. Explanations and practice were reviewed during development on 21 October 2026; Maurice should check their fit before assigning work.
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