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

The area of a circle of radius *r* is .

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
.

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 ,
the area *A* of the circle increases very slightly by an amount
approximately equal to (a ``ring'' around the circle of
length *C* and width ). So ,
and in the limit *dA*/*dr*=*C*.

Wed Apr 21 08:17:28 EDT 1999