In order to derive the formula for a circle, students must be familiar with the definitions of pi, diameter, radius, and the formula for circumference.

• Circumference: The distance around a circle (its perimeter)

• Diameter: A line segment that passes through the center of a circle with endpoints that terminate on the circle (diameter and length of diameter are used interchangeably)

• Radius: A line segment with one end point at the center of a circle and the other end point on the circle (radius and length of radius are used interchangeably)

• Pi: The ratio of the circumference of a circle to its diameter

It is also helpful to acknowledge that the radius of a circle is half its diameter and that the definition of pi naturally leads to the formula for the circumference C of a circle of diameter d (or radius r):

For our first step we take a circle and divide it into quarters and then dovetail them to form a kind of lumpy parallelogram with height r and a base length of approximately half the circumference. Obviously the area of the parallelogram gives only a rough estimate of the area of the four quarter circles.

So we repeat the process, this time dividing each quarter in half and dovetailing the smaller sectors to form another lumpy parallelogram with height r and a base length of approximately half the circumference. The area of this parallelogram gives an even better estimate of the eight sectors as the lumps are less pronounced.

Usually we can't do this (divide the circle into smaller and smaller equal sized wedges) with a physical model more than one or two more times but as you can see in the next figure, our parallelogram with height r and a base length of approximately half the circumference gives us an even better estimate of the area of our combined sectors. Notice also that the ends of our parallelogram are becoming more vertical with each iteration of this process.

Even though we can't keep cutting our wedges in half, if we imagine that we could do this infinitely many times, the resulting shape should be a rectangle with height r and base length equal to half the circumference of the circle since the arcs of half the circle align with the top of the rectangle and the other half align with the bottom.

In general, the area of a circle is the product of half the circumference C and the radius r:

Substituting the formula for circumfence in terms of the radius we obtain the familiar area formula:

Note: Most students readily agree that the formula for the area can be expressed in terms of the diameter of the circle but typically forget to account for the square when substituting the expression for radius r in terms of diameter d. Squaring half the diameter is equal to one fourth of the diameter squared and not one half the diameter squared. That is: