Pythagoras' Theory proved using hexagons

Pythagoras' Theory states that the square on the hypoteneuse of a right-angled triangle is equal to the sum of the squares on the other two sides.

This all sounds very square, and the Hall of Hexagons wishes to point out that hexagons can be used to prove the theory thus:-

We duplicate a hexagon; the area in each half is identical at each stage, until we find that the squares built on a and b together equal the square built on c (this is my own fun proof, rather similar to a proof usually ascribed to Leonardo).

For other proofs, click here

One way to look at the Pythagoras theory is to draw a line across a corner of a square:-

Pythagoras' theory says that:

Of course you'll want to know what happens when we draw a line across a corner of a hexagon:-

Instead of a right-angle in the usual Pythagoras triangle, we have 120 degrees. The formula is:-

For some reason this elegant result doesn't seem to be well known. It can be easily proved, knowing the formula for a right-angled triangle.

What happens when we draw a line across two corners of a hexagon?:-

By extending the sides of the hexagon (shown dotted) we have a 60 degree triangle, and the formula is:-

Another simple but often overlooked result. Proof here.

In fact, the general formula for a triangle with angle theta is:


120 degree triangle, with sides a, b and c:-

60 degree triangle, with sides a, b and c:-

page date: 23Oct05.      I enjoy correspondence stimulated by this site. You can contact me here.