Archimedes' used the exhaustion method for finding the area of a circle using a limiting process and a pair of polygons, one inscribed in the circle, one circumscribed around the circle, where the number of sides on the polygons is increased to provide better and better bounds on the area of the circle.
This applet allows you to choose circles of radii from 0.1 to 5. It also allows you to choose the number of sides [math]n[/math] on the polygons that are inscribed and circumscribed, anywhere from 3 sides to 30 sides. The areas of all three figures are then recorded so you can see both graphically and numerically how the area of the circle is better approximated as [math]n[/math] is increased.
