## Pi Filling, Archimedes Style!

9-12

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.

### Inscribed and Circumscribed Polygons

9-12
By calculating the areas of regular polygons inscribed and circumscribed about a unit circle, students create an algorithm that generates the never-ending digits of π, a common curiosity among high school students.

### Improving Archimedes' Method

9-12
Archimedes was the first mathematician to develop a converging series approximation to π. That highly influential discovery guided the development of calculus many hundreds of years later. However, his method only gives lower and upper boundaries that form intervals known to capture π, not a single numeric estimate of π. In this lesson, students ask, “Where is π located in those intervals?” They also discover an improvement to Archimedes' method that generates the infinite digits of π more efficiently and accurately.