# 三百六十角形

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## 正三百六十角形

${\displaystyle S={\frac {360}{4}}a^{2}\cot {\frac {\pi }{360}}\simeq 10312.97851a^{2}}$

${\displaystyle \sin {1^{\circ }}}$を平方根と立方根で表すと、

${\displaystyle \sin {1^{\circ }}={\frac {1+{\sqrt {3}}i}{16}}{\sqrt[{3}]{4{\sqrt {30}}-8{\sqrt {15+3{\sqrt {5}}}}+8{\sqrt {5+{\sqrt {5}}}}+4{\sqrt {10}}-4{\sqrt {6}}-4{\sqrt {2}}+\left(4{\sqrt {30}}+8{\sqrt {15+3{\sqrt {5}}}}+8{\sqrt {5+{\sqrt {5}}}}-4{\sqrt {10}}-4{\sqrt {6}}+4{\sqrt {2}}\right)i}}+}$

${\displaystyle \cos(2\pi /360)}$を平方根と立方根で表すと、

{\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{360}}=&{\sqrt[{3}]{\cos {\frac {2\pi }{120}}+i\sin {\frac {2\pi }{120}}}}+{\sqrt[{3}]{\cos {\frac {2\pi }{120}}-i\sin {\frac {2\pi }{120}}}}\\8\cos {\frac {2\pi }{360}}=&{\sqrt[{3}]{64\cos {\frac {2\pi }{120}}+i\cdot 64\sin {\frac {2\pi }{120}}}}+{\sqrt[{3}]{64\cos {\frac {2\pi }{120}}-i\cdot 64\sin {\frac {2\pi }{120}}}}\\\cos {\frac {2\pi }{360}}=&{\tfrac {{\sqrt[{3}]{4\left(2\left(1+{\sqrt {3}}\right){\sqrt {5+{\sqrt {5}}}}+\left({\sqrt {10}}-{\sqrt {2}}\right)\left({\sqrt {3}}-1\right)\right)+4\left(2\left(1-{\sqrt {3}}\right){\sqrt {5+{\sqrt {5}}}}+\left({\sqrt {10}}-{\sqrt {2}}\right)\left({\sqrt {3}}+1\right)\right)i}}+{\sqrt[{3}]{4\left(2\left(1+{\sqrt {3}}\right){\sqrt {5+{\sqrt {5}}}}+\left({\sqrt {10}}-{\sqrt {2}}\right)\left({\sqrt {3}}-1\right)\right)-4\left(2\left(1-{\sqrt {3}}\right){\sqrt {5+{\sqrt {5}}}}+\left({\sqrt {10}}-{\sqrt {2}}\right)\left({\sqrt {3}}+1\right)\right)i}}}{8}}\\\end{aligned}}}

## 脚注

 [脚注の使い方]