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:
Rearranging this, we get

If you know the diameter

The radius r of a circle is half the diameter d
Substituting r
Which simplifies to

If you know the circumference

The circumference c of a circle radius r is given by
Dividing both sides by 2π
Substitute this into the area formula for r
Which simplifies to