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The Pythagorean theorem explained simply
A simple explanation of the Pythagorean theorem (a² + b² = c²) with a visual proof, worked examples, Pythagorean triples and real-life uses. Free interactive lesson and worksheets included.
The Pythagorean theorem is one of the most useful ideas in geometry. It links the three sides of any right-angled triangle.
The rule
In a right-angled triangle, the longest side (opposite the right angle) is the hypotenuse, c. The other two sides are the legs, a and b. The theorem says:
a² + b² = c²
Why it works: squares on the sides
Draw a square on each side. The area of the square on the hypotenuse equals the sum of the areas of the squares on the two legs. For a 3-4-5 triangle: 9 + 16 = 25. Move the legs yourself in the interactive lesson The Pythagorean theorem.
Example 1: find the hypotenuse
Legs 6 and 8. c² = 36 + 64 = 100, so c = 10.
Example 2: find a leg
Hypotenuse 13, one leg 5. b² = 169 − 25 = 144, so b = 12.
Pythagorean triples
Sets of whole numbers that work: 3, 4, 5, 5, 12, 13, 8, 15, 17, 7, 24, 25. Any multiple works too (6, 8, 10).
Real-life uses
- Finding how long a ladder must be to reach a window
- The shortest distance between two points on a map or screen
- Checking a corner is square when building (3-4-5 rule)
- Computer graphics and games (distance between objects)
Common mistakes
- Using the theorem on a triangle with no right angle.
- Forgetting to take the square root at the end.
- Adding when finding a leg: for a leg, subtract.
Practise
Try the Pythagorean theorem worksheets with answer keys, or Maths Practice for instant feedback.
Common questions
Who discovered the Pythagorean theorem?
It is named after the Greek mathematician Pythagoras, but it was known to Babylonian and Indian mathematicians earlier. In India it appears in the Sulba Sutras.
What grade is the Pythagorean theorem taught in?
Usually around Grade 8 (Year 9, Class 8), ages 13 to 14.
Does it work for all triangles?
Only for right-angled triangles.
How do I know which side is the hypotenuse?
It is always the longest side, directly opposite the right angle.
This guide is general information. For exams, follow your school, teacher and official exam board, and use your own judgement about what suits your child.