Area of a circle  derivation
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 sixsided 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 into the area formula
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
While you are here..
... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone.
However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site?
When we reach the goal I will remove all advertising from the site.
It only takes a minute and any amount would be greatly appreciated.
Thank you for considering it! – John Page
Become a patron of the site at patreon.com/mathopenref
Other circle topics
General
Equations of a circle
Angles in a circle
Arcs
(C) 2011 Copyright Math Open Reference. All rights reserved
