The polygon can be broken down into n isosceles triangles (wheren is the number of sides), such as the one shown on the right.
In this triangle
s is the side length of the polygon
r is the radius of the polygon and the circle
h is the height of the triangle.
The area of the triangle is half the base times height orThere are n triangles in the polygon soThis can be rearranged to beThe term ns is the perimeter of the polygon (length of a side, times the number of sides). As the polygon gets to look more and more like a circle, this value approaches the circle circumference, which is 2πr. So, substituting 2πr for ns:Also, as the number of sides increases, the triangle gets narrower and narrower, and so when s approaches zero, h and r become the same length. So substituting r for h:Rearranging this, we get