IGCSE Maths Pythagoras' Theorem Notes
Pythagoras' theorem links the three sides of a right-angled triangle: the square of the hypotenuse equals the sum of the squares of the other two sides.
Key concepts
- Hypotenuse
- The longest side, opposite the right angle.
- Finding a shorter side
- Rearrange to subtract the squares: a² = c² − b².
- 3D Pythagoras
- Apply the theorem twice to find a diagonal through a solid.
Key formulas
Pythagoras
a² + b² = c²
c is the hypotenuse.
Worked example
A right-angled triangle has legs 6 cm and 8 cm. Find the hypotenuse.
- 1Square the legs: 6² + 8² = 36 + 64 = 100.
- 2Take the square root: √100 = 10.
- 3State the hypotenuse.
Answer: 10 cm
Common mistakes
✗ Adding when a shorter side is required.
✓ Subtract the squares to find a shorter side: a² = c² − b².
✗ Forgetting to square root at the end.
✓ The theorem gives the square — root it for the length.
✗ Using Pythagoras on a non-right triangle.
✓ It only works when there is a right angle.
Quick check
Try these, then reveal the answers.
Legs 3 and 4 — hypotenuse?Reveal answer
5
Hypotenuse 13, one leg 5 — other leg?Reveal answer
12
Is Pythagoras for right angles only?Reveal answer
Yes
Exam tips
- Identify the hypotenuse first — it is always opposite the right angle.
- Keep full accuracy until the final rounding step.
- For 3D, find a base diagonal first, then the sloping length.
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Frequently asked questions
How do I find a shorter side with Pythagoras?+
Rearrange to a² = c² − b², subtracting the squares, then take the square root.
Does Pythagoras work for any triangle?+
No — it applies only to right-angled triangles; use the sine or cosine rule otherwise.
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