Archimedes and the method of exhaustion

By Noah Veltman

If you establish that **π** is the ratio of a circle's circumference to its diameter, you can approximate the value of **π** using the "method of exhaustion" by inscribing a polygon inside the circle of radius **r** and then circumscribing a larger polygon outside the same circle.

The circumference of the circle (**c**) will lie somewhere between the perimeter of the smaller polygon (**p _{1}**) and the perimeter of the larger polygon (

As the polygon becomes many-sided, the estimate gets more and more accurate.

**Speed: **

sides

≤c≤

≤π≤