三十一角形
v 3 − 6 − x 1 + x 2 − x 3 12 v − ( 6 − x 1 + x 2 − x 3 ) ( 129 − 10 x 1 + 81 x 2 − 48 x 3 − 30 x 4 ) 216 ⋅ 67 = 0 {\displaystyle v^{3}-{\frac {6-x_{1}+x_{2}-x_{3}}{12}}v-{\frac {(6-x_{1}+x_{2}-x_{3})(129-10x_{1}+81x_{2}-48x_{3}-30x_{4})}{216\cdot 67}}=0}
三角関数、逆三角関数を用いた解は
cos  2 π 31 = x 1 6 + 1 3 6 − x 1 + x 2 − x 3 ⋅ cos  ( 1 3 arccos  129 − 10 x 1 + 81 x 2 − 48 x 3 − 30 x 4 134 6 − x 1 + x 2 − x 3 ) {\displaystyle \cos {\frac {2\pi }{31}}={\frac {x_{1}}{6}}+{\frac {1}{3}}{\sqrt {6-x_{1}+x_{2}-x_{3}}}\cdot \cos \left({\frac {1}{3}}\arccos {\frac {129-10x_{1}+81x_{2}-48x_{3}-30x_{4}}{134{\sqrt {6-x_{1}+x_{2}-x_{3}}}}}\right)}
平方根、立方根で表すと