十四角形

正十四角形においては、中心角と外角は25.714…°で、内角は154.285…°となる。一辺の長さが a の正十四角形の面積Sは

${\displaystyle S={\frac {14}{4}}a^{2}\cot {\frac {\pi }{14}}\simeq 15.3345a^{2}}$

となる。

${\displaystyle \cos(2\pi /14)}$を平方根と立方根で表すと

${\displaystyle \cos {\frac {2\pi }{14}}=\cos {\frac {\pi }{7}}={\frac {1}{6}}\left({\sqrt[{3}]{{\frac {7}{2}}\left(-1+3{\sqrt {3}}\cdot i\right)}}+{\sqrt[{3}]{{\frac {7}{2}}\left(-1-3{\sqrt {3}}\cdot i\right)}}+1\right)={\frac {1}{6}}\left({\sqrt {7}}\cdot {\sqrt[{3}]{\frac {-1+3{\sqrt {3}}\cdot i}{2{\sqrt {7}}}}}+{\sqrt {7}}\cdot {\sqrt[{3}]{\frac {-1-3{\sqrt {3}}\cdot i}{2{\sqrt {7}}}}}+1\right)=0.9009688...}$

Trigonometric constants expressed in real radicalsより

${\displaystyle \sin {\frac {2\pi }{14}}={\frac {\sqrt {3\left(28-2{\sqrt[{3}]{28-84i{\sqrt {3}}}}-2{\sqrt[{3}]{28+84i{\sqrt {3}}}}\right)}}{12}}}$

${\displaystyle \cos {\frac {2\pi }{14}}={\frac {\sqrt {3\left(20+2{\sqrt[{3}]{28-84i{\sqrt {3}}}}+2{\sqrt[{3}]{28+84i{\sqrt {3}}}}\right)}}{12}}}$

• Weisstein, Eric W. "Tetradecagon". mathworld.wolfram.com (英語).
• algebra precalculus - Roots of $8x^3-4x^2-4x+1$ - Mathematics Stack Exchange