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
S = 14 4 a 2 cot  π 14 ≃ 15.3345 a 2 {\displaystyle S={\frac {14}{4}}a^{2}\cot {\frac {\pi }{14}}\simeq 15.3345a^{2}}
となる。
cos  ( 2 π / 14 ) {\displaystyle \cos(2\pi /14)} を平方根と立方根で表すと
cos  2 π 14 = cos  π 7 = 1 6 ( 7 2 ( − 1 + 3 3 ⋅ i ) 3 + 7 2 ( − 1 − 3 3 ⋅ i ) 3 + 1 ) = 1 6 ( 7 ⋅ − 1 + 3 3 ⋅ i 2 7 3 + 7 ⋅ − 1 − 3 3 ⋅ i 2 7 3 + 1 ) = 0.9009688... {\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.