Pythagoras' theorem

This states that the area of a square drawn on the longest side of a right-angled triangle (the hypotenuse), will be equal to the sum of the areas of the squares drawn on the other two sides. The theorem is likely to have been known long before the time of Pythagoras. It was probably used by the ancient Egyptians to lay out the pyramids.

In geometry, a theorem stating that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. If the hypotenuse is h units long and the lengths of the other sides are a and b, then h² = a² + b².

The theorem provides a way of calculating the length of any side of a right-angled triangle if the lengths of the other two sides are known. For example, to find the length of a bridge constructed over a valley 120 m wide, when the vertical drop of the bridge is 20 m:

Using Pythagoras' theorem B² = 120² + 20², so B² = 14,400 + 400 = 14,800, giving b = √14,800 = 121.66 (to two decimal places).

The length of the bridge is 121.66 m.

Pythagoras' theorem is also used to determine certain trigonometric identities such as sin² θ + cos² θ = 1.