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.