and the perimeter of a polygon circumscribed outside a circle (which is greater than the circumference). The value of pi lies between those two lengths. By doubling the number of sides of the ...

The circumference of a circle is, in turn ... Archimedes used inscribed and circumscribed polygons such as these in his ...

Archimedes determined the upper and lower range of pi by finding the perimeters of inscribed and circumscribed polygons ... to the circumference of the circle and thereby come up with his ...