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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, then a 24-sided polygon, and finally 48- and 96-sided polygons, Archimedes was able to bring the two perimeters ever closer in length to the circumference of the circle and thereby come up with his approximation. Values are shown in decimal notation rather than the fractions that Archimedes used.

6-Sided Polygon inscribed perimeter = 3.0 circumscribed perimeter = 3.4641
12-Sided Polygon inscribed perimeter = 3.1058 circumscribed perimeter = 3.2154
24-Sided Polygon inscribed perimeter = 3.1326 circumscribed perimeter = 3.1597
48-Sided Polygon inscribed perimeter = 3.1394 circumscribed perimeter = 3.1461
96-Sided Polygon inscribed perimeter = 3.1410 circumscribed perimeter = 3.1427

actual value of [pi] = 3.1416

Note: All figures rounded off to four decimal places.