# 五十四角形

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## 正五十四角形

${\displaystyle S={\frac {27}{2}}a^{2}\cot {\frac {\pi }{54}}a^{2}}$

{\displaystyle {\begin{aligned}&\alpha =2\cos {\frac {2\pi }{54}}\cdot 2\cos {\frac {34\pi }{54}}\cdot 2\cos {\frac {38\pi }{54}}\\&\beta =2\cos {\frac {10\pi }{54}}\cdot 2\cos {\frac {46\pi }{54}}\cdot 2\cos {\frac {26\pi }{54}}\\&\gamma =2\cos {\frac {50\pi }{54}}\cdot 2\cos {\frac {14\pi }{54}}\cdot 2\cos {\frac {22\pi }{54}}\\\end{aligned}}}

{\displaystyle {\begin{aligned}&\alpha +\beta +\gamma =0\\&\alpha \beta +\beta \gamma +\gamma \alpha =-3\\&\alpha \beta \gamma =1\\\end{aligned}}}

${\displaystyle x^{3}-3x-1=0}$

${\displaystyle \alpha =2\cos \left({\frac {1}{3}}\arccos {\frac {1}{2}}\right)={\sqrt[{3}]{{\frac {1}{2}}+i{\frac {\sqrt {3}}{2}}}}+{\sqrt[{3}]{{\frac {1}{2}}-i{\frac {\sqrt {3}}{2}}}}={\sqrt[{3}]{-\omega ^{2}}}+{\sqrt[{3}]{-\omega }}}$

{\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{54}}+2\cos {\frac {34\pi }{54}}+2\cos {\frac {38\pi }{54}}=0\\&2\cos {\frac {2\pi }{54}}\cdot 2\cos {\frac {34\pi }{54}}+2\cos {\frac {34\pi }{54}}\cdot 2\cos {\frac {38\pi }{54}}+2\cos {\frac {38\pi }{54}}\cdot 2\cos {\frac {2\pi }{54}}=-3\\&2\cos {\frac {2\pi }{54}}\cdot 2\cos {\frac {34\pi }{54}}\cdot 2\cos {\frac {38\pi }{54}}=\alpha \\\end{aligned}}}

${\displaystyle u^{3}-3u-\alpha =0}$

${\displaystyle u_{1}=2\cos \left({\frac {1}{3}}\arccos {\frac {\alpha }{2}}\right)={\sqrt[{3}]{\sqrt[{3}]{-\omega ^{2}}}}+{\sqrt[{3}]{\sqrt[{3}]{-\omega }}}}$

よって

${\displaystyle \cos {\frac {2\pi }{54}}={\frac {{\sqrt[{3}]{\sqrt[{3}]{-\omega ^{2}}}}+{\sqrt[{3}]{\sqrt[{3}]{-\omega }}}}{2}}}$

## 脚注

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