Pythagoras’ theorem says that in a right-angled triangle, a² + b² = c², where c is the hypotenuse, the longest side opposite the right angle. To find the hypotenuse, add the squares of the shorter sides and square-root. To find a shorter side, subtract instead: a² = c² − b². The theorem also finds the distance between two points and checks whether a triangle is right-angled.
Pythagoras’ theorem
- a² + b² = c², where c is the hypotenuse
- Hypotenuse: c = √(a² + b²)
- Shorter side: a = √(c² − b²)
- Distance between points: √((x₂ − x₁)² + (y₂ − y₁)²)
The theorem
In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. The hypotenuse is always the longest side and always opposite the right angle, so identify it first.
Finding the hypotenuse
Square both shorter sides, add them, then take the square root. For sides 20 cm and 21 cm: 400 + 441 = 841, and √841 = 29 cm. Check that your answer is longer than both of the other sides.
Finding a shorter side
If you know the hypotenuse, subtract: a² = c² − b². For a hypotenuse of 41 cm and one side of 9 cm, a² = 1681 − 81 = 1600, so a = 40 cm. Your answer should be shorter than the hypotenuse; if it is longer, you have added instead of subtracting.
Pythagoras in real problems
Many problems hide a right-angled triangle: ladders against walls, diagonals of rectangles, and straight-line distances. Sketch the triangle, mark the right angle, decide whether the unknown is the hypotenuse, then add or subtract. A 7.5 m ladder with its foot 2.1 m from a wall reaches √(56.25 − 4.41) = 7.2 m up the wall.
Distances on a grid and checking right angles
The distance between two points is the hypotenuse of a triangle whose shorter sides are the horizontal and vertical differences. From (3, 1) to (12, 41), the differences are 9 and 40, so the distance is √(81 + 1600) = 41. To check whether a triangle is right-angled, see whether the two shorter sides squared add to the longest side squared.
Worked examples
Example 1. A right-angled triangle has shorter sides 20 cm and 21 cm. Find the hypotenuse.
- c² = 20² + 21² = 400 + 441 = 841.
- c = √841.
Answer: 29 cm.
Example 2. A right-angled triangle has hypotenuse 41 cm and one shorter side 9 cm. Find the other side.
- a² = 41² − 9² = 1681 − 81 = 1600.
- a = √1600.
Answer: 40 cm.
Example 3. A right-angled triangle has shorter sides 7 cm and 9 cm. Find the hypotenuse to 2 decimal places.
- c² = 49 + 81 = 130.
- c = √130 = 11.401…
Answer: 11.40 cm.
Example 4. A 7.5 m ladder leans against a wall with its foot 2.1 m from the wall. How high up the wall does it reach?
- The ladder is the hypotenuse, so subtract.
- h² = 7.5² − 2.1² = 56.25 − 4.41 = 51.84.
- h = √51.84.
Answer: 7.2 m.
Example 5. Find the distance between the points (3, 1) and (12, 41).
- Horizontal difference: 9. Vertical difference: 40.
- √(9² + 40²) = √1681.
Answer: 41 units.
Common mistakes
- Adding when finding a shorter side. Subtract: a² = c² − b².
- Forgetting to square-root. c² = 841 means c = 29, not 841.
- Using the wrong side as the hypotenuse. The hypotenuse is opposite the right angle.
- Squaring the sum. (20 + 21)² is not 20² + 21².
- Using Pythagoras without a right angle. The theorem only applies to right-angled triangles.
How Pythagoras’ theorem is examined
Pythagoras’ theorem is examined at Foundation and Higher on GCSE (1MA1) and iGCSE (4MA1) papers alike, from simple triangle questions to 3D problems at Higher. Questions ask you to:
- find the hypotenuse of a right-angled triangle
- find a shorter side
- solve a problem involving ladders, diagonals or distances
- find the distance between two coordinate points
The Pythagoras worksheet includes full answers, and the geometry engine creates new triangles every time.
Pythagoras is used with trigonometry, and it underpins surd answers and the distance formula at A-Level.
Frequently asked questions
What is Pythagoras’ theorem?
In a right-angled triangle, a² + b² = c², where c is the hypotenuse.
How do you find a shorter side?
Subtract the square of the known shorter side from the square of the hypotenuse, then square-root.
How do you know if a triangle is right-angled?
Check whether the squares of the two shorter sides add up to the square of the longest side.