This example shows how to derive the area of circle formula using calculus. There are multiple ways to derive this formula, this method involves taking the integral of a function that models area in terms of the radius of the circle.

The setup for deriving this formula involves viewing the area of a the circle as a function of the radius. Abstractly, this can be visualized by sub-dividing the area into cocentric rings. As the number of rings approaches infinity the area of the rings converges on the area of the circle. For example, shown below are some cases of dividing the area of the circle into rings.

The area of of these rings can be set up as an integral. The function we are integrating over returns the circumference of the circle given its radius. This is illustrated below by the thin band of area highlighted green.

The area of this band is given by its length (circumference) multiplied by its width (radius). Recall that the circumference of a circle is given by the formula . This is the function we are integrating over and since we are integrating with respect to the variable , the width of the band “” is the differential of the integral.

Set up the integral of the function starting from to , where is the radius of the circle and little is the current radius of the function.

Factor out the circle constant from the integral.

Take the integral of the function.

Evaluate the definite integral.

Simplify the expression.

Finally change the big back to the little to represent the radius of the circle.

This gives us the area of circle formula.

The area of a circle is give by one-half multiplied by τ (tau) mutliplied by the radius of the circle squared.