# 六十五角形

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## 正六十五角形

${\displaystyle S={\frac {65}{4}}a^{2}\cot {\frac {\pi }{65}}\simeq 335.95298a^{2}}$

{\displaystyle {\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{65}}+2\cos {\frac {32\pi }{65}}+2\cos {\frac {8\pi }{65}}={\frac {{\frac {{\frac {1+{\sqrt {13}}}{2}}+{\sqrt {\frac {35+{\sqrt {325}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65+{\sqrt {2925}}}{2}}-{\sqrt {\frac {715+{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\&x_{2}=2\cos {\frac {4\pi }{65}}+2\cos {\frac {64\pi }{65}}+2\cos {\frac {16\pi }{65}}={\frac {{\frac {{\frac {1-{\sqrt {13}}}{2}}+{\sqrt {\frac {35-{\sqrt {325}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65-{\sqrt {2925}}}{2}}+{\sqrt {\frac {715-{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\&x_{3}=2\cos {\frac {6\pi }{65}}+2\cos {\frac {34\pi }{65}}+2\cos {\frac {24\pi }{65}}={\frac {{\frac {{\frac {1+{\sqrt {13}}}{2}}-{\sqrt {\frac {35+{\sqrt {325}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65+{\sqrt {2925}}}{2}}+{\sqrt {\frac {715+{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\&x_{4}=2\cos {\frac {12\pi }{65}}+2\cos {\frac {62\pi }{65}}+2\cos {\frac {48\pi }{65}}={\frac {{\frac {{\frac {1-{\sqrt {13}}}{2}}-{\sqrt {\frac {35-{\sqrt {325}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65-{\sqrt {2925}}}{2}}-{\sqrt {\frac {715-{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\&x_{5}=2\cos {\frac {18\pi }{65}}+2\cos {\frac {28\pi }{65}}+2\cos {\frac {58\pi }{65}}={\frac {{\frac {{\frac {1+{\sqrt {13}}}{2}}+{\sqrt {\frac {35+{\sqrt {325}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65+{\sqrt {2925}}}{2}}-{\sqrt {\frac {715+{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\&x_{6}=2\cos {\frac {36\pi }{65}}+2\cos {\frac {56\pi }{65}}+2\cos {\frac {14\pi }{65}}={\frac {{\frac {{\frac {1-{\sqrt {13}}}{2}}+{\sqrt {\frac {35-{\sqrt {325}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65-{\sqrt {2925}}}{2}}+{\sqrt {\frac {715-{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\&x_{7}=2\cos {\frac {54\pi }{65}}+2\cos {\frac {46\pi }{65}}+2\cos {\frac {44\pi }{65}}={\frac {{\frac {{\frac {1+{\sqrt {13}}}{2}}-{\sqrt {\frac {35+{\sqrt {325}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65+{\sqrt {2925}}}{2}}+{\sqrt {\frac {715+{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\&x_{8}=2\cos {\frac {22\pi }{65}}+2\cos {\frac {38\pi }{65}}+2\cos {\frac {42\pi }{65}}={\frac {{\frac {{\frac {1-{\sqrt {13}}}{2}}-{\sqrt {\frac {35-{\sqrt {325}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65-{\sqrt {2925}}}{2}}-{\sqrt {\frac {715-{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\\end{aligned}}}

{\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{65}}\cdot 2\cos {\frac {32\pi }{65}}+2\cos {\frac {32\pi }{65}}\cdot 2\cos {\frac {8\pi }{65}}+2\cos {\frac {8\pi }{65}}\cdot 2\cos {\frac {2\pi }{65}}\\&=x_{3}+2\cos {\frac {2\pi }{13}}+2\cos {\frac {6\pi }{13}}+2\cos {\frac {8\pi }{13}}\\&=x_{3}+{\frac {-1+{\sqrt {13}}}{2}}\\&2\cos {\frac {2\pi }{65}}\cdot 2\cos {\frac {32\pi }{65}}\cdot 2\cos {\frac {8\pi }{65}}=x_{8}+2\cos {\frac {2\pi }{5}}=x_{8}+{\frac {-1+{\sqrt {5}}}{2}}\\\end{aligned}}}

${\displaystyle u^{3}-x_{1}u^{2}+(x_{3}+{\frac {-1+{\sqrt {13}}}{2}})u-(x_{8}+{\frac {-1+{\sqrt {5}}}{2}})=0}$

## 脚注

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