Archimedes (287–212 BC) realized that the magnitude of pi can be bounded from below and above by inscribing circles in regular polygons and calculating the outer and inner polygons' respective perimeters.

By using the equivalent of 96-sided polygons, he proved that 3 10/71 < π < 3 1/7.

The average of these values is about 3.1418511.

By using the equivalent of 96-sided polygons, he proved that 3 10/71 < π < 3 1/7.

The average of these values is about 3.1418511.