五十二角形
正五十二角形においては、中心角と外角は6.923076...°で、内角は173.076923...°となる。一辺の長さが a の正五十二角形の面積 S は
S = 13 a 2 cot  π 52 a 2 {\displaystyle S=13a^{2}\cot {\frac {\pi }{52}}a^{2}}
x 1 = 2 cos  2 π 52 + 2 cos  18 π 52 + 2 cos  46 π 52 = 13 − 3 13 2 x 2 = 2 cos  10 π 52 + 2 cos  14 π 52 + 2 cos  22 π 52 = 13 + 3 13 2 x 3 = 2 cos  50 π 52 + 2 cos  34 π 52 + 2 cos  6 π 52 = − 13 − 3 13 2 x 4 = 2 cos  42 π 52 + 2 cos  38 π 52 + 2 cos  30 π 52 = − 13 + 3 13 2 {\displaystyle {\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{52}}+2\cos {\frac {18\pi }{52}}+2\cos {\frac {46\pi }{52}}={\sqrt {\frac {13-3{\sqrt {13}}}{2}}}\\&x_{2}=2\cos {\frac {10\pi }{52}}+2\cos {\frac {14\pi }{52}}+2\cos {\frac {22\pi }{52}}={\sqrt {\frac {13+3{\sqrt {13}}}{2}}}\\&x_{3}=2\cos {\frac {50\pi }{52}}+2\cos {\frac {34\pi }{52}}+2\cos {\frac {6\pi }{52}}=-{\sqrt {\frac {13-3{\sqrt {13}}}{2}}}\\&x_{4}=2\cos {\frac {42\pi }{52}}+2\cos {\frac {38\pi }{52}}+2\cos {\frac {30\pi }{52}}=-{\sqrt {\frac {13+3{\sqrt {13}}}{2}}}\\\end{aligned}}}