## Pi Filling, Archimedes Style!

In the spirit of Archimedes’ method of approximating pi, students inscribe and circumscribe regular polygons in and around the unit circle, which is known to have an area of π.

Students then consider the area of
the polygons, using either an applet or a graphing calculator. The area of the *n*‑gon
will approach π as *n*
increases.

Similarly, students consider the perimeter of inscribed and circumscribed regular polygons in and around a circle with unit diameter. This exploration also leads to an approximation of π.

Taken together, the two methods provide a compelling investigation of a method for generating the never‑ending digits of π.

**Prerequisite
Knowledge**

Prior to this lesson, students should have a
solid knowledge of plane geometry; they should be familiar with the sine,
cosine, and tangent functions, and be able to use them to find the side lengths
of triangles; and they should be able to use the formula (1/2)*ap*, which can be
used to find the area of a triangle with apothem *a* and perimeter *p*.