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Deriving the Area Formula for a Circle


Date: 03/08/2000 at 17:56:38
From: Richard Atkinson
Subject: Area of a circle as it relates to pi

Why is the area of a circle the square of the radius times pi?


Date: 03/09/2000 at 03:23:03
From: Doctor Floor
Subject: Re: Area of a circle as it relates to pi

Hi, Richard,

Thanks for writing.

Let's consider a circle with radius r.

If we divide the circle into an even number of sectors, we can 
rearrange these sectors as in the following figure:



The result is a sort of wrongly formed rectangle, but we know that the 
shorter "side" of this rectangle has length r, and that the longer 
"side" is half the perimeter, hence pi*r.

The more sectors we make, the more accurate our rectangle becomes. We 
can imagine that if we divided the circle into an infinite number of 
sectors, it would become a rectangle.

Whatever the number of sectors we use, the "side" lengths will remain 
r and pi*r. Therefore our limit case with an infinite number of 
sectors still has sides r and pi*r, and the area of this limit 
rectangle is pi*r^2. Since the area of the circle does not change when 
we divide it into parts, the area of the circle must have been pi*r^2, 
too.

If you need more help, just write back.

Best regards,

- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
Elementary Circles
Elementary Geometry
Elementary Two-Dimensional Geometry
Middle School Conic Sections/Circles
Middle School Geometry
Middle School Two-Dimensional Geometry

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