Area of a Circle 2.
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.
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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.
Bettina.Richmond@wku.edu
Updated May 22, 2008.