Why is Area of a Circle Equal to Pi * (Radius Squared)?

Date: 07/05/2005 at 07:45:01
From: Sandra
Subject: why does pi r square give the area 

I know the formula for the area of a circle is A = pi*r^2, but why
does that give the area?

Date: 07/05/2005 at 08:52:52
From: Doctor Jerry
Subject: Re: why does pi r square give the area

Hello Sandra,

One answer is "Because one can prove that the area of a circle of
radius r is pi*r^2, using calculus."  

There are other ways, not of proving that the area of a circle of
radius r is pi*r^2, but of persuading others that this is the right
formula.

Kepler gave a nice, persuasive argument.  It depends upon your 
accepting that the circumference of a circle of radius r is 2*pi*r. 
See the diagram at

   

Divide the region K enclosed by a circle by rays from the origin to
points of the circle, doing the division so that there are n congruent
sectors.  The central angle of each sector is 2*pi/n.  

Now imagine that we cut the region R along the radii forming the
boundaries of the sectors, but do not cut into the circle itself.  We
then imagine "unrolling" this pie along its edge, so that the edge
becomes a straight line and the sectors are upright, as on the left in 
the figure.  Each sector is very much like a triangle.  The area of
one of these triangles is one-half times its base times its height:

  (1/2)*(2*pi*r/n)*(r)  

Because there are n "triangles," their total area, which is the same
as area of the circle, is

  n*(1/2)*(2*pi*r/n)*(r),
 
which is pi*r^2.  

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/