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 (p1) and the perimeter of the larger polygon (p2). Therefore the value of π is somewhere between p1/2r and p2/2r.

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