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The regular 8-sided Polygon. The Inradius $r$, Circumradius $R$, and Area $A$ can be computed directly from the formulas for a general regular Polygon with side length $s$ and $n=8$ sides,

$\displaystyle r$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}s\cot\left({\pi\over 8}\right)={\textstyle{1\over 2}}(1+\sqrt{2}\,)s$ (1)
$\displaystyle R$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}s\csc\left({\pi\over 8}\right)={\textstyle{1\over 2}}\sqrt{4+2\sqrt{2}}\,s$ (2)
$\displaystyle A$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}ns^2\cot\left({\pi\over 8}\right)=2(1+\sqrt{2}\,)s^2.$ (3)

See also Octahedron, Polygon, Regular Polygon, Trigonometry Values Pi/8

© 1996-9 Eric W. Weisstein