七十二角形

正七十二角形においては、中心角と外角は5°で、内角は175°となる。一辺の長さが a の正七十二角形の面積 S は

${\displaystyle S=18a^{2}\cot {\frac {\pi }{72}}}$

${\displaystyle \sin(2\pi /72)}$を平方根と立方根で表すと、

${\displaystyle \sin {\frac {2\pi }{72}}={\frac {2-2{\sqrt {3}}\mathrm {i} }{2{\sqrt[{3}]{2({\sqrt {2}}-{\sqrt {6}})}}-2-{\sqrt {3}}}}-{\frac {(1+{\sqrt {3}}\mathrm {i} ){\sqrt[{3}]{2({\sqrt {2}}-{\sqrt {6}})}}-2-{\sqrt {3}}}{8}}\,}$

関係式

${\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{72}}+2\cos {\frac {50\pi }{72}}+2\cos {\frac {46\pi }{72}}=0\\2\cos {\frac {14\pi }{72}}+2\cos {\frac {62\pi }{72}}+2\cos {\frac {34\pi }{72}}=0\\2\cos {\frac {10\pi }{72}}+2\cos {\frac {38\pi }{72}}+2\cos {\frac {58\pi }{72}}=0\\2\cos {\frac {70\pi }{72}}+2\cos {\frac {22\pi }{72}}+2\cos {\frac {26\pi }{72}}=0\\\end{aligned}}}$

三次方程式の係数を求めると

${\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{72}}\cdot 2\cos {\frac {50\pi }{72}}+2\cos {\frac {50\pi }{72}}\cdot 2\cos {\frac {46\pi }{72}}+2\cos {\frac {46\pi }{72}}\cdot 2\cos {\frac {2\pi }{72}}=-3\\&2\cos {\frac {2\pi }{72}}\cdot 2\cos {\frac {50\pi }{72}}\cdot 2\cos {\frac {46\pi }{72}}=2\cos {\frac {2\pi }{24}}\end{aligned}}}$

解と係数の関係より

${\displaystyle u^{3}-3u-2\cos {\frac {2\pi }{24}}=0}$

三次方程式を解いて、整理すると${\displaystyle \cos(2\pi /72)}$が求められる。

${\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{72}}=&{\sqrt[{3}]{\cos {\frac {2\pi }{24}}+i\sin {\frac {2\pi }{24}}}}+{\sqrt[{3}]{\cos {\frac {2\pi }{24}}-i\sin {\frac {2\pi }{24}}}}\\4\cos {\frac {2\pi }{72}}=&{\sqrt[{3}]{8\cos {\frac {2\pi }{24}}+i8\sin {\frac {2\pi }{24}}}}+{\sqrt[{3}]{8\cos {\frac {2\pi }{24}}-i8\sin {\frac {2\pi }{24}}}}\\4\cos {\frac {2\pi }{72}}=&{\sqrt[{3}]{2({\sqrt {6}}+{\sqrt {2}})+i\cdot 2({\sqrt {6}}-{\sqrt {2}})}}+{\sqrt[{3}]{2({\sqrt {6}}+{\sqrt {2}})-i\cdot 2({\sqrt {6}}-{\sqrt {2}})}}\\\end{aligned}}}$

${\displaystyle \cos {\frac {2\pi }{72}}={\frac {1}{4}}{\sqrt[{3}]{2({\sqrt {6}}+{\sqrt {2}})+i\cdot 2({\sqrt {6}}-{\sqrt {2}})}}+{\frac {1}{4}}{\sqrt[{3}]{2({\sqrt {6}}+{\sqrt {2}})-i\cdot 2({\sqrt {6}}-{\sqrt {2}})}}}$

• 九角形
• 十二角形
• 十八角形
• 二十四角形
• 三十六角形