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Approximating Pi
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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.


inscribed polygons   circumscribed polygons

6-Sided Polygon
inscribed perimeter = 3.0
circumscribed perimeter = 3.4641



inscribed polygons   circumscribed polygons

12-Sided Polygon
inscribed perimeter = 3.1058
circumscribed perimeter = 3.2154



inscribed polygons   circumscribed polygons

24-Sided Polygon
inscribed perimeter = 3.1326
circumscribed perimeter = 3.1597



inscribed polygons   circumscribed polygons

48-Sided Polygon
inscribed perimeter = 3.1394
circumscribed perimeter = 3.1461



inscribed polygons   circumscribed polygons

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.

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