Area of a Circle 1.

The formula for the area of a circle is a little more difficult and requires methods of successively better approximations.

 The usual definition of π is the ratio of the circumference of a circle to its diameter d, so that the circumference of a circle is πd, or  2πr, where r is the radius.

 The animation below shows that a circle can be cut and rearranged to closely resemble a parallelogram (with height  r and base πr) of area πr ². By dividing the circle into more than eight slices, the approximation obtained in this manner would be even better. By dividing the circle into more and more slices, the approximating parallelograms approximate the area of the circle arbitrarily closely. This gives a geometric justification that the area of a circle really is πr ².

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 Rectangle :  Parallelogram :  Triangle 1. :  Triangle 2.  :  Trapezoid 1.  :  Trapezoid 2. :  Circle 1. :  Circle 2.

Animations by Bettina Richmond and Tom Richmond. 
Tom.Richmond@wku.edu
Updated May 22, 2008.