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.