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Proof of Pythagoras Theorem is taught in the note for Pythagoras Theorem. Read it below and look for the section on this topic.

Pythagoras Theorem

8 min read

The theorem

In a right-angled triangle the longest side, opposite the right angle, is the hypotenuse. Pythagoras' theorem says the square on the hypotenuse equals the sum of the squares on the other two sides.

c is the hypotenuse; a and b are the two shorter sides.
The areas of the two small squares add up to the area of the big one.
Pythagoras’ theoremA right-angled triangle with sides 3, 4 and 5. A square is drawn on each side: the squares on the two shorter sides have areas 9 and 16, which add up to the area 25 of the square on the longest side, the hypotenuse.b² = 16a² = 9c² = 25345a² + b² = c²9 + 16 = 25

Using it

Try itFind the hypotenuse

Choose the two shorter sides. The bars compare the three squares: the first two together equal the third.

Worked exampleFind the hypotenuse

A ladder leans on a wall. Its foot is 2 m from the wall and it reaches 5 m up. How long is the ladder (to 1 decimal place)?

  1. The wall, ground and ladder form a right-angled triangle. The ladder is the hypotenuse.
  2. Apply the theorem.
  3. Square root.

Answer: The ladder is about 5.4 m long.

Worked exampleFind a shorter side

The hypotenuse is 13 cm and one side is 5 cm. Find the other side.

  1. Rearrange: .
  2. Square root.

Answer: The other side is 12 cm.

Check yourself A right-angled triangle has sides 6 cm and 8 cm. What is the hypotenuse?
Check yourself Could a triangle with sides 7, 9 and 12 have a right angle?

Key terms

Hypotenuse
The longest side of a right-angled triangle, opposite the right angle.
Pythagorean triple
Three whole numbers a, b, c with a² + b² = c².

Summary

  • In a right-angled triangle a² + b² = c².
  • Find the hypotenuse by adding squares and taking the square root.
  • Find a shorter side by subtracting squares.
  • If a² + b² = c² the triangle is right-angled.
Application of Pythagoras Theorem