Archimedes determined the upper and lower range of pi by finding the perimeters of inscribed and circumscribed polygons. By doubling the number of sides of the hexagon to a 12-sided polygon ...

By doubling the number of sides of the hexagon to a 12-sided polygon, then a 24-sided polygon, and finally 48- and 96-sided polygons, Archimedes was able to bring the two perimeters ever closer in ...