六十五角形
x 1 = 2 cos  2 π 65 + 2 cos  32 π 65 + 2 cos  8 π 65 = 1 + 13 2 + 35 + 325 2 2 + 65 + 2925 2 − 715 + 494325 2 2 2 x 2 = 2 cos  4 π 65 + 2 cos  64 π 65 + 2 cos  16 π 65 = 1 − 13 2 + 35 − 325 2 2 + 65 − 2925 2 + 715 − 494325 2 2 2 x 3 = 2 cos  6 π 65 + 2 cos  34 π 65 + 2 cos  24 π 65 = 1 + 13 2 − 35 + 325 2 2 + 65 + 2925 2 + 715 + 494325 2 2 2 x 4 = 2 cos  12 π 65 + 2 cos  62 π 65 + 2 cos  48 π 65 = 1 − 13 2 − 35 − 325 2 2 − 65 − 2925 2 − 715 − 494325 2 2 2 x 5 = 2 cos  18 π 65 + 2 cos  28 π 65 + 2 cos  58 π 65 = 1 + 13 2 + 35 + 325 2 2 − 65 + 2925 2 − 715 + 494325 2 2 2 x 6 = 2 cos  36 π 65 + 2 cos  56 π 65 + 2 cos  14 π 65 = 1 − 13 2 + 35 − 325 2 2 − 65 − 2925 2 + 715 − 494325 2 2 2 x 7 = 2 cos  54 π 65 + 2 cos  46 π 65 + 2 cos  44 π 65 = 1 + 13 2 − 35 + 325 2 2 − 65 + 2925 2 + 715 + 494325 2 2 2 x 8 = 2 cos  22 π 65 + 2 cos  38 π 65 + 2 cos  42 π 65 = 1 − 13 2 − 35 − 325 2 2 + 65 − 2925 2 − 715 − 494325 2 2 2 {\displaystyle {\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{65}}+2\cos {\frac {32\pi }{65}}+2\cos {\frac {8\pi }{65}}={\frac {{\frac {{\frac {1+{\sqrt {13}}}{2}}+{\sqrt {\frac {35+{\sqrt {325}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65+{\sqrt {2925}}}{2}}-{\sqrt {\frac {715+{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\&x_{2}=2\cos {\frac {4\pi }{65}}+2\cos {\frac {64\pi }{65}}+2\cos {\frac {16\pi }{65}}={\frac {{\frac {{\frac {1-{\sqrt {13}}}{2}}+{\sqrt {\frac {35-{\sqrt {325}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65-{\sqrt {2925}}}{2}}+{\sqrt {\frac {715-{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\&x_{3}=2\cos {\frac {6\pi }{65}}+2\cos {\frac {34\pi }{65}}+2\cos {\frac {24\pi }{65}}={\frac {{\frac {{\frac {1+{\sqrt {13}}}{2}}-{\sqrt {\frac {35+{\sqrt {325}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65+{\sqrt {2925}}}{2}}+{\sqrt {\frac {715+{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\&x_{4}=2\cos {\frac {12\pi }{65}}+2\cos {\frac {62\pi }{65}}+2\cos {\frac {48\pi }{65}}={\frac {{\frac {{\frac {1-{\sqrt {13}}}{2}}-{\sqrt {\frac {35-{\sqrt {325}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65-{\sqrt {2925}}}{2}}-{\sqrt {\frac {715-{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\&x_{5}=2\cos {\frac {18\pi }{65}}+2\cos {\frac {28\pi }{65}}+2\cos {\frac {58\pi }{65}}={\frac {{\frac {{\frac {1+{\sqrt {13}}}{2}}+{\sqrt {\frac {35+{\sqrt {325}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65+{\sqrt {2925}}}{2}}-{\sqrt {\frac {715+{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\&x_{6}=2\cos {\frac {36\pi }{65}}+2\cos {\frac {56\pi }{65}}+2\cos {\frac {14\pi }{65}}={\frac {{\frac {{\frac {1-{\sqrt {13}}}{2}}+{\sqrt {\frac {35-{\sqrt {325}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65-{\sqrt {2925}}}{2}}+{\sqrt {\frac {715-{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\&x_{7}=2\cos {\frac {54\pi }{65}}+2\cos {\frac {46\pi }{65}}+2\cos {\frac {44\pi }{65}}={\frac {{\frac {{\frac {1+{\sqrt {13}}}{2}}-{\sqrt {\frac {35+{\sqrt {325}}}{2}}}}{2}}-{\sqrt {\frac {{\frac {65+{\sqrt {2925}}}{2}}+{\sqrt {\frac {715+{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\&x_{8}=2\cos {\frac {22\pi }{65}}+2\cos {\frac {38\pi }{65}}+2\cos {\frac {42\pi }{65}}={\frac {{\frac {{\frac {1-{\sqrt {13}}}{2}}-{\sqrt {\frac {35-{\sqrt {325}}}{2}}}}{2}}+{\sqrt {\frac {{\frac {65-{\sqrt {2925}}}{2}}-{\sqrt {\frac {715-{\sqrt {494325}}}{2}}}}{2}}}}{2}}\\\end{aligned}}}