Deriving the Area Formula for a CircleDate: 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/ |
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