Pythagoras

For a right angled triangle whose shorter sides are of length a and b, and whose longer side (hypotenuse) is of length c, Pythagoras' theorem states a2 + b2 = c2.

This page demonstrates geometric constructs to demonstrate the theorem. They comprise four triangles each of area a*b/2 (hence having a combined area 2ab), occupying some or all of a square of size a+b.

In the first construct, the four triangles surround a square of size c. This shows (a+b)2 = 2ab + c2.

In the second construct, shown two ways, they surround a square of size b-a, forming a square of size c.

The first way shows c2 = 2ab + (b-a)2.

The second way, where two of the triangles are moved to form squares of size a and b, shows c2 = a2 + b2.

Pythagorean triples are sets of three integers a, b and c which fulfill the theorem. The page draws the constructs with some of these triples. Also given are inverse triples. See also my Inverse Pythagoras page

Use the 'explain' buttons to find out more.

The first method is widely used, the second was explained by J Bronowski in Ascent of Man, episode 5, who stated that this was probably how Pythagoras demonstrated his theorem (as it does not need algebra).

However, triangles can be moved with the first method too - see my Pythagoras and Binomial page

Like WS Gilbert's Major General said, there are many useful facts about the square on the hypotenuse!