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Q.E.D.

8 9 From PIe to PI mysteries of the circle Eratosthenes 276 194 B.C.E. is famous for his pisza pie method for calculating the circumference of the Earth based on the distance from Alexandria to Syene and the shadow angle at Alexandria at a time when the sun shone down a deep well in Syene, throwing no shadow there. Using the formula circle diameter . p circle circumference, he also calculated the Earths diameter. Fortunately, his penpal Archimedes 287 212 B.C.E. was able to prove a good estimate for the elusive value of p. Since p is the circumference of a circle of diameter 1, it will be greater than the circumference of any inscribed and less than that of any circumscribed regular polygon opposite, top. The more sides such a polygon has, the closer its circumference will be to that of the circle. Luckily, it is easy to calculate from the circumference of one such polygon the circumference of a polygon of the same kind with double the number of sides opposite, middle. Starting with regular hexagons, Archimedes successively calculated the circumferences of regular 12, 24, 48 and 96gons to capture p between 3 and 3 . The last of these values equals and is used even today in many school books instead of the true value of p. Using squares instead of hexagons, a formula for approximating p emerges lower, opposite.
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