二百五十五角形
cos  2 π 255 = cos  ( π 15 − π 17 ) = cos  π 15 cos  π 17 + sin  π 15 sin  π 17 = 1 8 ( − 1 + 5 + 30 + 180 ) ⋅ cos  π 17 + 1 8 ( + 3 − 15 + 10 + 20 ) ⋅ sin  π 17 = 1 8 ( − 1 + 5 + 30 + 180 ) ⋅ 1 16 ( + 1 − 17 + 34 − 68 + 68 + 2448 + 2720 + 6284288 ) + 1 8 ( + 3 − 15 + 10 + 20 ) ⋅ 1 8 ( 34 − 68 − 136 − 1088 − 272 + 39168 − 43520 + 1608777728 ) {\displaystyle {\begin{aligned}\cos {\frac {2\pi }{255}}=&\cos \left({\frac {\pi }{15}}-{\frac {\pi }{17}}\right)\\&=\cos {\frac {\pi }{15}}\cos {\frac {\pi }{17}}+\sin {\frac {\pi }{15}}\sin {\frac {\pi }{17}}\\&={\frac {1}{8}}\left({-1+{\sqrt {5}}+{\sqrt {30+{\sqrt {180}}}}}\right)\cdot \cos {\frac {\pi }{17}}+{\frac {1}{8}}\left({+{\sqrt {3}}-{\sqrt {15}}+{\sqrt {10+{\sqrt {20}}}}}\right)\cdot \sin {\frac {\pi }{17}}\\&={\frac {1}{8}}\left({-1+{\sqrt {5}}+{\sqrt {30+{\sqrt {180}}}}}\right)\cdot {\frac {1}{16}}\left({+1-{\sqrt {17}}+{\sqrt {34-{\sqrt {68}}}}+{\sqrt {68+{\sqrt {2448}}+{\sqrt {2720+{\sqrt {6284288}}}}}}}\right)+{\frac {1}{8}}\left({+{\sqrt {3}}-{\sqrt {15}}+{\sqrt {10+{\sqrt {20}}}}}\right)\cdot {\frac {1}{8}}\left({\sqrt {34-{\sqrt {68}}-{\sqrt {136-{\sqrt {1088}}}}-{\sqrt {272+{\sqrt {39168}}-{\sqrt {43520+{\sqrt {1608777728}}}}}}}}\right)\\\end{aligned}}}