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 ...

It finds an approximation of pi by determining the length of the perimeter of a polygon inscribed within a circle (which is less than the circumference of the circle) and the perimeter of a ...

Archimedes used inscribed and circumscribed polygons such as these in his approximation of pi. Credit: Public domain Ancient ...