# Pi (Polygons based) Calculator

## Calculates circular constant Pi using the perimeters of regular polygons inscribed in and circumscribed about a circle of diameter 1.

 initial regular polygon tetragon (4) hexagon (6) loop frequency n calculated up to  2 or 3 x 2n -sided polygon
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 The calculation stops when the perimeters of both circumscribed and inscribed regular polygons become equal. The accuracy of π improves by increasing the number of digits for calculation.From ancient times until the 17th century, the approximation of Pi was calculated from the perimeters of the circumscribed and inscribed regular polygons.Pi is calculated from the perimeters of from the initial tetragon or hexagon to the 2 or 3 x 2n-sided circumscribed and inscribed regular polygons. Iterative algorithms　a:perimeter of circumscribed polygon, b:perimeter of inscribed polygon$\normal(1)\ a_0=\left\{{\large{4\atop 2\sqrt{3}}}\right.\ ,\hspace{20}b_0=\left\{{\large{2\sqrt{2}\hspace{20}:square\atop\hspace{20} 3\hspace{30}:hexagon}}\right.\\(2)\ a_{n+1}={\large\frac{2a_n b_n}{a_n+b_n}}\ ,\hspace{20}b_{n+1}=sqrt{a_{n+1}b_n}\\\vspace{5}(3)\ b_n\lt\ \pi\ \lt a_n\\$

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