# Pythagoras Revisited

In Euclidean geometry, one of the most well-known theorems is the Pythagorean theorem. It states that, in a right-angle triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

For a right triangle with hypotenuse

*c*, and other sides*a, b*the equation states:There are dozens of ways to prove the theorem, but a simple one is a geometric rearrangement.

Starting with a line of length

*a+b*, we can make four copies of this to make a square.Connecting the points

*a*away from the corners makes another square, internally, with side*c*. This also forms four right-angle triangles.This composition can be rearranged to make two rectangles of area

*ab*and the square*c*. The two configurations are the same total area. A little bit of algebra confirms the theorem.^{2}Advertisement:

## Areas

You've probably seen the formula represented graphically by placing squares on each edge of the triangle. As the area of each square is the edge length squared, there is a simple relationship between the areas of the squares.

There's even some fabulous physical demonstrations of this, as seen in this video:

## Other Shapes

What might blow your mind is that it also works for Pentagons.

Or triangles …

Or any similar shape …

## Why?

The answer is pretty simple. When you scale up any shape a linear dimension, its area scales up by the

*square*of that change.It seems obvious now, when you think about it, but the first time seen it is a surprise to many.

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