Maths
Pythagoras' theorem — quick study summary
In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It works in reverse too — if a² + b² = c², the triangle is right-angled.
Key points
- Only applies to right-angled triangles
- Hypotenuse (c) is always opposite the right angle and is the longest side
- Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25)…
- Used for distance between two points in coordinate geometry
In more depth
Finding a missing side
Label the hypotenuse c: it is the longest side, opposite the right angle. To find the hypotenuse, square the other two sides, add them and take the square root. To find a shorter side, square the hypotenuse, subtract the square of the known shorter side and take the square root. Check that the hypotenuse comes out longest.
Pythagorean triples and the converse
Whole-number sets such as 3, 4, 5 and 5, 12, 13 satisfy a² + b² = c², as do their multiples, such as 6, 8, 10. The converse also holds: if a triangle's sides satisfy a² + b² = c², the triangle has a right angle.
Coordinates and 3D
The distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²), which is Pythagoras on the coordinate grid. In a cuboid with sides a, b and c, the space diagonal is √(a² + b² + c²), found by applying the theorem twice.
Common mistakes
- Adding the squares when finding a shorter side. Subtract from the hypotenuse squared instead.
- Using the theorem on triangles that are not right-angled.
- Forgetting the final square root and giving c² as the answer.
Practice quiz
Click each question to reveal the answer.
1. A right triangle has legs of length 6 and 8. What is the hypotenuse?
- 7
- 10
- 14
- 100
Answer: 10
√(36+64) = √100 = 10.
2. If c = 13 and one leg is 5, what is the other leg?
Answer: 12
√(169−25) = √144 = 12.
3. Does (3, 4, 5) form a right triangle?
Answer: Yes
9 + 16 = 25, so a² + b² = c².
Last reviewed: May 2026