Derivation of the formula for the area of a circle

This page describes how to derive the formula for the area of a circle. we start with a regular polygon and show that as the number of sides gets very large, the figure becomes a circle. By finding the area of the polygon we derive the equation for the area of a circle.

Try this. In the applet below we have a six-sided regular polygon. Keep clicking on 'more' and note that as the number of sides gets larger, the polygon approaches being a circle. As the number of sides becomes infinitely large, it is, in fact, a circle. Click 'reset' afterwards.

The polygon can be broken down into n isosceles triangles (where n 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 or

There are n triangles in the polygon so

This can be rearranged to be

The 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: