四十二角形
正四十二角形においては、中心角と外角は8.571...°で、内角は171.428...°となる。一辺の長さが a の正四十二角形の面積 S は
S = 42 4 a 2 cot  π 42 ≃ 140.11276 a 2 {\displaystyle S={\frac {42}{4}}a^{2}\cot {\frac {\pi }{42}}\simeq 140.11276a^{2}}
cos  ( 2 π / 42 ) {\displaystyle \cos(2\pi /42)} を平方根と立方根で表すことが可能である。
cos  2 π 42 = cos  π 21 = 1 12 72 + 72 cos  2 π 21 = 1 12 72 + 72 ⋅ 1 + 21 + 154 − 30 21 + ( 42 3 − 18 7 ) i 3 + 154 − 30 21 + ( 18 7 − 42 3 ) i 3 12 = 1 12 72 + 6 ( 1 + 21 + 154 − 30 21 + ( 42 3 − 18 7 ) i 3 + 154 − 30 21 + ( 18 7 − 42 3 ) i 3 ) {\displaystyle {\begin{aligned}\cos {\frac {2\pi }{42}}=&\cos {\frac {\pi }{21}}\\=&{\frac {1}{12}}{\sqrt {72+72\cos {\frac {2\pi }{21}}}}\\=&{\frac {1}{12}}{\sqrt {72+72\cdot {\frac {1+{\sqrt {21}}+{\sqrt[{3}]{154-30{\sqrt {21}}+\left(42{\sqrt {3}}-18{\sqrt {7}}\right)i}}+{\sqrt[{3}]{154-30{\sqrt {21}}+\left(18{\sqrt {7}}-42{\sqrt {3}}\right)i}}}{12}}}}\\=&{\frac {1}{12}}{\sqrt {72+6\left({1+{\sqrt {21}}+{\sqrt[{3}]{154-30{\sqrt {21}}+\left(42{\sqrt {3}}-18{\sqrt {7}}\right)i}}+{\sqrt[{3}]{154-30{\sqrt {21}}+\left(18{\sqrt {7}}-42{\sqrt {3}}\right)i}}}\right)}}\end{aligned}}}