Area of a Circle or Regular Polygon

There's a nice way to see why the formula for the area of a circle of radius R is: pi * R².
It has an interesting relationship with the formula for the circumference of a circle, which is 2 * Pi * R (and that is a consequence of the definition of Pi, which is defined to be the ratio of the circumference of a circle to its diameter.)

So consider a regular polygon, which is an N-sided figure with equal side lengths S and equal angles at each corner. There is an inscribed circle to the polygon that has center C and just barely touches the midpoint of every side. A line from C to the midpoint of a side is called the apothem, and suppose this apothem has length R.

If you cut the polygon along lines from each corner of the polygon to the center C, you will get a bunch of triangles, each with area (1/2)*(base)*(height). Note that each (base) has length S and the (height) is the length R of the apothem, and there are N such triangles. Thus the total area of the polygon is N*(1/2)*S*R, which to say it another way is:
(1/2) (Circumference of the Polygon) * R

Now notice that if you let N, the number of sides of the polygon, get larger and larger, the polygon's area approaches the area of a circle of radius R. On the other hand, the circumference approaches the circumference of a circle, so that as N goes to infinity, the above formula approaches: 
(1/2) (2 * Pi * R) * R

which is just Pi*R², the area of a circle!

Presentation Suggestions:
Draw a circle and cut it into thin pie wedges. Then help students see that each pie wedge is approximated by a very thin triangle, and as we cut the pie into more and more wedges, this approximation gets better and better and in the limit the approximation becomes equality.

The Math Behind the Fact:
The process of approximating the area of the circle by slicing it into thin wedges is analogous to the process of integration (in calculus) to find an area. In the limit this approximation gets better and better. To see the relationship between circumference and area in reverse, where derivatives play a role, see Surface Area of a sphere. See Rolling Polygons for more connections between polygons and circles.

How to Cite this Page: 
Su, Francis E., et al. “Area of a Circle or Regular Polygon.” Math Fun Facts.

Fun Fact suggested by:
Francis Su