二十三角形

正二十三角形においては、中心角と外角は15.652…°で、内角は164.347…°となる。一辺の長さが a の正二十三角形の面積 S は

${\displaystyle S={\frac {23}{4}}a^{2}\cot {\frac {\pi }{23}}\simeq 41.83436a^{2}}$

${\displaystyle \cos(2\pi /23)}$の値は、11次方程式を解くことにより冪根で表現される[1]。${\displaystyle z^{11}=1}$の複素数解の一つ ${\displaystyle e^{{\frac {2\pi }{11}}i}}$ をσとおいて、10次多項式にσを代入した値の11乗根を10個(${\displaystyle \lambda _{1},\lambda _{2},\lambda _{3},\lambda _{4},\lambda _{5},\lambda _{6},\lambda _{7},\lambda _{8},\lambda _{9},\lambda _{10}}$)用いて表される。

${\displaystyle \cos {\frac {2\pi }{23}}={\frac {\lambda _{1}+\lambda _{2}+\lambda _{3}+\lambda _{4}+\lambda _{5}+\lambda _{6}+\lambda _{7}+\lambda _{8}+\lambda _{9}+\lambda _{10}-1}{22}}}$

${\displaystyle \lambda _{1}={\sqrt[{11}]{23(384812+188298\sigma -625515\sigma ^{2}-78859\sigma ^{3}+740707\sigma ^{4}+84370\sigma ^{5}+834405\sigma ^{6}+98208\sigma ^{7}+361900\sigma ^{8}-56177\sigma ^{9})}}}$

${\displaystyle \lambda _{2}={\sqrt[{11}]{23(384812+188298\sigma ^{2}-625515\sigma ^{4}-78859\sigma ^{6}+740707\sigma ^{8}+84370\sigma ^{10}+834405\sigma +98208\sigma ^{3}+361900\sigma ^{5}-56177\sigma ^{7})}}}$

${\displaystyle \lambda _{3}={\sqrt[{11}]{23(384812+188298\sigma ^{3}-625515\sigma ^{6}-78859\sigma ^{9}+740707\sigma +84370\sigma ^{4}+834405\sigma ^{7}+98208\sigma ^{10}+361900\sigma ^{2}-56177\sigma ^{5})}}}$

${\displaystyle \lambda _{4}={\sqrt[{11}]{23(384812+188298\sigma ^{4}-625515\sigma ^{8}-78859\sigma +740707\sigma ^{5}+84370\sigma ^{9}+834405\sigma ^{2}+98208\sigma ^{6}+361900\sigma ^{10}-56177\sigma ^{3})}}}$

${\displaystyle \lambda _{5}={\sqrt[{11}]{23(384812+188298\sigma ^{5}-625515\sigma ^{10}-78859\sigma ^{4}+740707\sigma ^{9}+84370\sigma ^{3}+834405\sigma ^{8}+98208\sigma ^{2}+361900\sigma ^{7}-56177\sigma )}}}$

${\displaystyle \lambda _{6}={\sqrt[{11}]{23(384812+188298\sigma ^{6}-625515\sigma -78859\sigma ^{7}+740707\sigma ^{2}+84370\sigma ^{8}+834405\sigma ^{3}+98208\sigma ^{9}+361900\sigma ^{4}-56177\sigma ^{10})}}}$

${\displaystyle \lambda _{7}={\sqrt[{11}]{23(384812+188298\sigma ^{7}-625515\sigma ^{3}-78859\sigma ^{10}+740707\sigma ^{6}+84370\sigma ^{2}+834405\sigma ^{9}+98208\sigma ^{5}+361900\sigma -56177\sigma ^{8})}}}$

${\displaystyle \lambda _{8}={\sqrt[{11}]{23(384812+188298\sigma ^{8}-625515\sigma ^{5}-78859\sigma ^{2}+740707\sigma ^{10}+84370\sigma ^{7}+834405\sigma ^{4}+98208\sigma +361900\sigma ^{9}-56177\sigma ^{6})}}}$

${\displaystyle \lambda _{9}={\sqrt[{11}]{23(384812+188298\sigma ^{9}-625515\sigma ^{7}-78859\sigma ^{5}+740707\sigma ^{3}+84370\sigma +834405\sigma ^{10}+98208\sigma ^{8}+361900\sigma ^{6}-56177\sigma ^{4})}}}$

${\displaystyle \lambda _{10}={\sqrt[{11}]{23(384812+188298\sigma ^{10}-625515\sigma ^{9}-78859\sigma ^{8}+740707\sigma ^{7}+84370\sigma ^{6}+834405\sigma ^{5}+98208\sigma ^{4}+361900\sigma ^{3}-56177\sigma ^{2})}}}$

• 1の冪根
• 十一角形

Weisstein, Eric W. "Trigonometry angles". mathworld.wolfram.com (英語).

• π/23