How the Area of a Disk Grows
An increase in the radius of a circular disk leads to a corresponding increase in the area of the disk. Geometrically, is the red annulus. Slicing and then unrolling this annulus would yield a shape close to a rectangle with base and height . This gives the approximation . Using the slider, notice this approximation is quite accurate for close to zero. Graphing the area as a function of the radius , the ratio is the slope of a secant line to the curve. Dividing both sides of by implies the slope may be approximated by the circumference .
Taking the limit as approaches zero, the approximation becomes , which expresses the derivative of with respect to .