四十一角形
( 2 cos  2 π 41 + 2 cos  18 π 41 ) + ( 2 cos  6 π 41 + 2 cos  28 π 41 ) = x 1 ( 2 cos  4 π 41 + 2 cos  36 π 41 ) + ( 2 cos  12 π 41 + 2 cos  26 π 41 ) = x 2 ( 2 cos  8 π 41 + 2 cos  10 π 41 ) + ( 2 cos  24 π 41 + 2 cos  30 π 41 ) = x 3 ( 2 cos  16 π 41 + 2 cos  20 π 41 ) + ( 2 cos  22 π 41 + 2 cos  34 π 41 ) = x 4 ( 2 cos  32 π 41 + 2 cos  40 π 41 ) + ( 2 cos  14 π 41 + 2 cos  38 π 41 ) = x 5 {\displaystyle {\begin{aligned}&\left(2\cos {\frac {2\pi }{41}}+2\cos {\frac {18\pi }{41}}\right)+\left(2\cos {\frac {6\pi }{41}}+2\cos {\frac {28\pi }{41}}\right)=x_{1}\\&\left(2\cos {\frac {4\pi }{41}}+2\cos {\frac {36\pi }{41}}\right)+\left(2\cos {\frac {12\pi }{41}}+2\cos {\frac {26\pi }{41}}\right)=x_{2}\\&\left(2\cos {\frac {8\pi }{41}}+2\cos {\frac {10\pi }{41}}\right)+\left(2\cos {\frac {24\pi }{41}}+2\cos {\frac {30\pi }{41}}\right)=x_{3}\\&\left(2\cos {\frac {16\pi }{41}}+2\cos {\frac {20\pi }{41}}\right)+\left(2\cos {\frac {22\pi }{41}}+2\cos {\frac {34\pi }{41}}\right)=x_{4}\\&\left(2\cos {\frac {32\pi }{41}}+2\cos {\frac {40\pi }{41}}\right)+\left(2\cos {\frac {14\pi }{41}}+2\cos {\frac {38\pi }{41}}\right)=x_{5}\\\end{aligned}}}
和積の公式より
正四十一角形においては、中心角と外角は8.78...°で、内角は171.219...°となる。一辺の長さが a の正四十一角形の面積 S は
S = 41 4 a 2 cot  π 41 ≃ 133.50783 a 2 {\displaystyle S={\frac {41}{4}}a^{2}\cot {\frac {\pi }{41}}\simeq 133.50783a^{2}}
( 2 cos  8 π 41 ⋅ 2 cos  10 π 41 ) + ( 2 cos  24 π 41 ⋅ 2 cos  30 π 41 ) = x 1 ( 2 cos  16 π 41 ⋅ 2 cos  20 π 41 ) + ( 2 cos  22 π 41 ⋅ 2 cos  34 π 41 ) = x 2 ( 2 cos  32 π 41 ⋅ 2 cos  40 π 41 ) + ( 2 cos  14 π 41 ⋅ 2 cos  38 π 41 ) = x 3 ( 2 cos  2 π 41 ⋅ 2 cos  18 π 41 ) + ( 2 cos  6 π 41 ⋅ 2 cos  28 π 41 ) = x 4 ( 2 cos  4 π 41 ⋅ 2 cos  36 π 41 ) + ( 2 cos  12 π 41 ⋅ 2 cos  26 π 41 ) = x 5 {\displaystyle {\begin{aligned}&\left(2\cos {\frac {8\pi }{41}}\cdot 2\cos {\frac {10\pi }{41}}\right)+\left(2\cos {\frac {24\pi }{41}}\cdot 2\cos {\frac {30\pi }{41}}\right)=x_{1}\\&\left(2\cos {\frac {16\pi }{41}}\cdot 2\cos {\frac {20\pi }{41}}\right)+\left(2\cos {\frac {22\pi }{41}}\cdot 2\cos {\frac {34\pi }{41}}\right)=x_{2}\\&\left(2\cos {\frac {32\pi }{41}}\cdot 2\cos {\frac {40\pi }{41}}\right)+\left(2\cos {\frac {14\pi }{41}}\cdot 2\cos {\frac {38\pi }{41}}\right)=x_{3}\\&\left(2\cos {\frac {2\pi }{41}}\cdot 2\cos {\frac {18\pi }{41}}\right)+\left(2\cos {\frac {6\pi }{41}}\cdot 2\cos {\frac {28\pi }{41}}\right)=x_{4}\\&\left(2\cos {\frac {4\pi }{41}}\cdot 2\cos {\frac {36\pi }{41}}\right)+\left(2\cos {\frac {12\pi }{41}}\cdot 2\cos {\frac {26\pi }{41}}\right)=x_{5}\\\end{aligned}}}