四十五角形
正四十五角形においては、中心角と外角は8°で、内角は172°となる。一辺の長さが a の正四十五角形の面積 S は
S = 45 4 a 2 cot  π 45 ≃ 160.8825 a 2 {\displaystyle S={\frac {45}{4}}a^{2}\cot {\frac {\pi }{45}}\simeq 160.8825a^{2}}
cos  ( 2 π / 45 ) {\displaystyle \cos(2\pi /45)} を平方根と立方根で表すと、
cos  2 π 45 = cos  ( π 9 − π 15 ) = cos  π 9 cos  π 15 + sin  π 9 sin  π 15 = 4 − 4 i 3 3 + 4 + 4 i 3 3 4 ⋅ 1 8 ( 6 ( 5 + 5 ) + 5 − 1 ) + i ( 4 − 4 i 3 3 − 4 + 4 i 3 3 ) 4 ⋅ 1 8 ( 2 ( 5 + 5 ) + 3 − 15 ) = 1 32 ( ( 4 − 4 i 3 3 + 4 + 4 i 3 3 ) ( 6 ( 5 + 5 ) + 5 − 1 ) + i ( 4 − 4 i 3 3 − 4 + 4 i 3 3 ) ( 2 ( 5 + 5 ) + 3 − 15 ) ) {\displaystyle {\begin{aligned}\cos {\frac {2\pi }{45}}=&\cos \left({\frac {\pi }{9}}-{\frac {\pi }{15}}\right)\\=&\cos {\frac {\pi }{9}}\cos {\frac {\pi }{15}}+\sin {\frac {\pi }{9}}\sin {\frac {\pi }{15}}\\=&{\frac {{\sqrt[{3}]{4-4i{\sqrt {3}}}}+{\sqrt[{3}]{4+4i{\sqrt {3}}}}}{4}}\cdot {\frac {1}{8}}\left({\sqrt {6\left(5+{\sqrt {5}}\right)}}+{\sqrt {5}}-1\right)+{\frac {i\left({\sqrt[{3}]{4-4i{\sqrt {3}}}}-{\sqrt[{3}]{4+4i{\sqrt {3}}}}\right)}{4}}\cdot {\frac {1}{8}}\left({\sqrt {2\left(5+{\sqrt {5}}\right)}}+{\sqrt {3}}-{\sqrt {15}}\right)\\=&{\frac {1}{32}}\left(\left({{\sqrt[{3}]{4-4i{\sqrt {3}}}}+{\sqrt[{3}]{4+4i{\sqrt {3}}}}}\right)\left({\sqrt {6\left(5+{\sqrt {5}}\right)}}+{\sqrt {5}}-1\right)+{i\left({\sqrt[{3}]{4-4i{\sqrt {3}}}}-{\sqrt[{3}]{4+4i{\sqrt {3}}}}\right)}\left({\sqrt {2\left(5+{\sqrt {5}}\right)}}+{\sqrt {3}}-{\sqrt {15}}\right)\right)\end{aligned}}}