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 divided into concentric rings which can be unrolled to closely resemble a triangle (with height r and base 2πr) of area πr ². By dividing the circle into more rings and unrolling, the result more closely matches a triangle, so the approximation becomes better. Taking more and more rings, the triangles approximate the area of the circle arbitrarily closely and we conclude that the area of a circle is πr ².
Rectangle : Parallelogram : Triangle
1. : Triangle
2. : Trapezoid 1.
1. : Circle
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Bettina Richmond and Tom Richmond.
Updated May 22, 2008.