九十七角形

正九十七角形においては、中心角と外角は3.711…°で、内角は176.288…°となる。一辺の長さが a の正九十七角形の面積 S は

${\displaystyle S={\frac {97}{4}}a^{2}\cot {\frac {\pi }{97}}\simeq 748.48261a^{2}}$

関係式

${\displaystyle {\begin{aligned}x_{1}=&2\cos {\frac {2\pi }{97}}+2\cos {\frac {72\pi }{97}}+2\cos {\frac {70\pi }{97}}\\x_{2}=&2\cos {\frac {10\pi }{97}}+2\cos {\frac {28\pi }{97}}+2\cos {\frac {38\pi }{97}}\\x_{3}=&2\cos {\frac {50\pi }{97}}+2\cos {\frac {54\pi }{97}}+2\cos {\frac {4\pi }{97}}\\x_{4}=&2\cos {\frac {56\pi }{97}}+2\cos {\frac {76\pi }{97}}+2\cos {\frac {20\pi }{97}}\\x_{5}=&2\cos {\frac {86\pi }{97}}+2\cos {\frac {8\pi }{97}}+2\cos {\frac {94\pi }{97}}\\x_{6}=&2\cos {\frac {42\pi }{97}}+2\cos {\frac {40\pi }{97}}+2\cos {\frac {82\pi }{97}}\\x_{7}=&2\cos {\frac {16\pi }{97}}+2\cos {\frac {6\pi }{97}}+2\cos {\frac {22\pi }{97}}\\x_{8}=&2\cos {\frac {80\pi }{97}}+2\cos {\frac {30\pi }{97}}+2\cos {\frac {84\pi }{97}}\\x_{9}=&2\cos {\frac {12\pi }{97}}+2\cos {\frac {44\pi }{97}}+2\cos {\frac {32\pi }{97}}\\x_{10}=&2\cos {\frac {60\pi }{97}}+2\cos {\frac {26\pi }{97}}+2\cos {\frac {34\pi }{97}}\\x_{11}=&2\cos {\frac {88\pi }{97}}+2\cos {\frac {64\pi }{97}}+2\cos {\frac {24\pi }{97}}\\x_{12}=&2\cos {\frac {52\pi }{97}}+2\cos {\frac {68\pi }{97}}+2\cos {\frac {74\pi }{97}}\\x_{13}=&2\cos {\frac {66\pi }{97}}+2\cos {\frac {48\pi }{97}}+2\cos {\frac {18\pi }{97}}\\x_{14}=&2\cos {\frac {58\pi }{97}}+2\cos {\frac {46\pi }{97}}+2\cos {\frac {90\pi }{97}}\\x_{15}=&2\cos {\frac {96\pi }{97}}+2\cos {\frac {36\pi }{97}}+2\cos {\frac {62\pi }{97}}\\x_{16}=&2\cos {\frac {92\pi }{97}}+2\cos {\frac {14\pi }{97}}+2\cos {\frac {78\pi }{97}}\\\end{aligned}}}$

${\displaystyle (x_{1}+x_{9}),\quad (x_{1}-x_{9})^{2}}$のように和、差の二乗を計算すると

${\displaystyle {\begin{aligned}x_{1}=&{\frac {{\frac {{\frac {{\frac {-1+{\sqrt {97}}}{2}}-{\sqrt {\frac {97-{\sqrt {7857}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {97-{\sqrt {2425}}}{2}}+{\sqrt {\frac {5141-{\sqrt {25927033}}}{2}}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {{\frac {97+{\sqrt {4753}}}{2}}-{\sqrt {\frac {1649+{\sqrt {2705233}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {2037+{\sqrt {3841297}}}{2}}-{\sqrt {\frac {3488605+{\sqrt {12162451874697}}}{2}}}}{2}}}}{2}}}}{2}}\\=&{\frac {{\frac {{\frac {{\frac {-1+{\sqrt {97}}}{2}}-{\sqrt {\frac {97-9{\sqrt {97}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {97-5{\sqrt {97}}}{2}}+{\sqrt {\frac {5141-517{\sqrt {97}}}{2}}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {{\frac {97+7{\sqrt {97}}}{2}}-{\sqrt {\frac {1649+167{\sqrt {97}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {2037+199{\sqrt {97}}}{2}}-{\sqrt {\frac {3488605+354099{\sqrt {97}}}{2}}}}{2}}}}{2}}}}{2}}\\\end{aligned}}}$

三次方程式の係数を求めると

${\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{97}}\cdot 2\cos {\frac {72\pi }{97}}+2\cos {\frac {72\pi }{97}}\cdot 2\cos {\frac {70\pi }{97}}+2\cos {\frac {70\pi }{97}}\cdot 2\cos {\frac {2\pi }{97}}=x_{1}+x_{12}\\&2\cos {\frac {2\pi }{97}}\cdot 2\cos {\frac {72\pi }{97}}\cdot 2\cos {\frac {70\pi }{97}}=x_{3}+2\\\end{aligned}}}$

解と係数の関係より

${\displaystyle u^{3}-x_{1}u^{2}+(x_{1}+x_{12})u-(x_{3}+2)=0}$

この三次方程式を解くことにより、${\displaystyle \cos(2\pi /97)}$を平方根と立方根で表すことが可能である。

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