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For centuries, mathematicians around the world have struggled in the chase for an increasingly accurate approximation of pi. The first recorded avenue of approximating the value of pi by theoretical means was the polygon approximation method, employed by Archimedes around 250 B.C., which was accurate only to about two decimal places when he performed it (Groleau). This was partly due to the fact that Greek mathematicians at the time had no concept of 0, thus only used fractions, not decimals, and the fact that they had no access to algebra (McKeeman). Now, over 2000 years later, pi has been calculated and verified to more than 12.1 trillion digits thanks, in part, to the Bailey-Borwein-Plouffe (BBP) formula, published in 1996, which allows us to quickly and easily calculate a specific digit of pi (Yee and Kondo; Seife). Though the polygon method and the BBP formula each work in entirely*…show more content…*

The polygon approximation method uses the perimeters of polygons to approximate pi and “was the first theoretical, rather than measured, calculation of pi,” but while it worked great for Archimedes in his time, it has limited practicality today (Groleau). In order to estimate pi, Archimedes used a circle that had a polygon inscribed in and circumscribed about it, then found the perimeter of each polygon, and used those values as the upper and lower limits of pi (Groleau). He started with a hexagon as the inner polygon of a circle with a diameter of 1 whose perimeter is equal to 6r when r is the radius of the circle (McKeeman). This meant that since the circumference of the circle was 2πr, 6r < 2πr and the lower limit of pi was 3 (McKeeman). He then found the perimeter of the outer hexagon to calculate the upper limit and proceeded to double the number of sides and repeat the

The polygon approximation method uses the perimeters of polygons to approximate pi and “was the first theoretical, rather than measured, calculation of pi,” but while it worked great for Archimedes in his time, it has limited practicality today (Groleau). In order to estimate pi, Archimedes used a circle that had a polygon inscribed in and circumscribed about it, then found the perimeter of each polygon, and used those values as the upper and lower limits of pi (Groleau). He started with a hexagon as the inner polygon of a circle with a diameter of 1 whose perimeter is equal to 6r when r is the radius of the circle (McKeeman). This meant that since the circumference of the circle was 2πr, 6r < 2πr and the lower limit of pi was 3 (McKeeman). He then found the perimeter of the outer hexagon to calculate the upper limit and proceeded to double the number of sides and repeat the

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## Circumference Formula

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