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Area and Circumference of Circles

From the above we know that the circumference C of a circle of radius r is tex2html_wrap_inline629 .

The area of a circle of radius r is tex2html_wrap_inline633 .

This can be proved by calculus or ``seen'' by cutting up a circle into many, very small, equally-sized, pie-shaped pieces which can then be rearranged into a figure which is approximately a parallelogram of height r and length C/2. Its area is therefore rC/2 which equals tex2html_wrap_inline641 .

Why is the formula for the circumference equal to the derivative of the formula for the area? Because when r is increased very slightly by an amount tex2html_wrap_inline645 , the area A of the circle increases very slightly by an amount tex2html_wrap_inline649 approximately equal to tex2html_wrap_inline651 (a ``ring'' around the circle of length C and width tex2html_wrap_inline645 ). So tex2html_wrap_inline657 , and in the limit dA/dr=C.

Carl Lee
Wed Apr 21 08:17:28 EDT 1999