三十五角形（さんじゅうごかくけい、さんじゅうごかっけい、triacontapentagon）は、多角形の一つで、35本の辺と35個の頂点を持つ図形である。内角の和は5940°、対角線の本数は560本である。
正三十五角形
正三十五角形においては、中心角と外角は10.285…°で、内角は169.714…°となる。一辺の長さが a の正三十五角形の面積 S は
を平方根と立方根で表すことが可能である。
![{\displaystyle {\begin{aligned}\cos {\frac {2\pi }{35}}=&\cos \left({\frac {\pi }{5}}-{\frac {\pi }{7}}\right)\\=&\cos {\frac {\pi }{5}}\cos {\frac {\pi }{7}}+\sin {\frac {\pi }{5}}\sin {\frac {\pi }{7}}\\=&{\frac {1}{4}}\left({\sqrt {5}}+1\right)\cdot \cos {\frac {\pi }{7}}+{\frac {1}{4}}\left({\sqrt {10-2{\sqrt {5}}}}\right)\cdot \sin {\frac {\pi }{7}}\\=&{\frac {{\sqrt {5}}+1}{4}}\cdot {\frac {\sqrt {3\left(20+2{\sqrt[{3}]{28-84i{\sqrt {3}}}}+2{\sqrt[{3}]{28+84i{\sqrt {3}}}}\right)}}{12}}+{\frac {\sqrt {10-2{\sqrt {5}}}}{4}}\cdot {\frac {\sqrt {3\left(28-2{\sqrt[{3}]{28-84i{\sqrt {3}}}}-2{\sqrt[{3}]{28+84i{\sqrt {3}}}}\right)}}{12}}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/7c4a402c598ab50453ff9646ee89eda962dafe88)
- 関係式
さらに、以下のような関係式が得られる。
両辺の立方根を取ると
![{\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{35}}+\omega \cdot 2\cos {\frac {32\pi }{35}}+\omega ^{2}\cdot 2\cos {\frac {22\pi }{35}}=&{\sqrt[{3}]{\tfrac {{-7+49{\sqrt {5}}+15{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}}-3{\sqrt {3}}\left(7+7{\sqrt {5}}+{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)i}{8}}}\\2\cos {\frac {2\pi }{35}}+\omega ^{2}\cdot 2\cos {\frac {32\pi }{35}}+\omega \cdot 2\cos {\frac {22\pi }{35}}=&{\sqrt[{3}]{\tfrac {{-7+49{\sqrt {5}}+15{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}}+3{\sqrt {3}}\left(7+7{\sqrt {5}}+{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)i}{8}}}\\\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a5d13d480cf9626407d8d014295e0e955154251c)
よって
![{\displaystyle {\begin{aligned}\cos {\frac {2\pi }{35}}=&{\frac {1}{6}}\left({\tfrac {1+{\sqrt {5}}-{\sqrt {14\left(5-{\sqrt {5}}\right)}}}{4}}+{\sqrt[{3}]{\tfrac {{-7+49{\sqrt {5}}+15{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}}-3{\sqrt {3}}\left(7+7{\sqrt {5}}+{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)i}{8}}}+{\sqrt[{3}]{\tfrac {{-7+49{\sqrt {5}}+15{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}}+3{\sqrt {3}}\left(7+7{\sqrt {5}}+{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)i}{8}}}\right)\\\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/6b13afa575f6e8963e6d411b6eeb8a3e1b8f3073)
正三十五角形の作図
正三十五角形は定規とコンパスによる作図が不可能な図形である。
正三十五角形は折紙により作図可能である。
![{\displaystyle {\begin{aligned}\cos {\frac {2\pi }{35}}=&\cos \left({\frac {\pi }{5}}-{\frac {\pi }{7}}\right)\\=&\cos {\frac {\pi }{5}}\cos {\frac {\pi }{7}}+\sin {\frac {\pi }{5}}\sin {\frac {\pi }{7}}\\=&{\frac {1}{4}}\left({\sqrt {5}}+1\right)\cdot \cos {\frac {\pi }{7}}+{\frac {1}{4}}\left({\sqrt {10-2{\sqrt {5}}}}\right)\cdot \sin {\frac {\pi }{7}}\\=&{\frac {{\sqrt {5}}+1}{4}}\cdot {\frac {\sqrt {3\left(20+2{\sqrt[{3}]{28-84i{\sqrt {3}}}}+2{\sqrt[{3}]{28+84i{\sqrt {3}}}}\right)}}{12}}+{\frac {\sqrt {10-2{\sqrt {5}}}}{4}}\cdot {\frac {\sqrt {3\left(28-2{\sqrt[{3}]{28-84i{\sqrt {3}}}}-2{\sqrt[{3}]{28+84i{\sqrt {3}}}}\right)}}{12}}\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/7c4a402c598ab50453ff9646ee89eda962dafe88)
![{\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{35}}+\omega \cdot 2\cos {\frac {32\pi }{35}}+\omega ^{2}\cdot 2\cos {\frac {22\pi }{35}}=&{\sqrt[{3}]{\tfrac {{-7+49{\sqrt {5}}+15{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}}-3{\sqrt {3}}\left(7+7{\sqrt {5}}+{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)i}{8}}}\\2\cos {\frac {2\pi }{35}}+\omega ^{2}\cdot 2\cos {\frac {32\pi }{35}}+\omega \cdot 2\cos {\frac {22\pi }{35}}=&{\sqrt[{3}]{\tfrac {{-7+49{\sqrt {5}}+15{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}}+3{\sqrt {3}}\left(7+7{\sqrt {5}}+{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)i}{8}}}\\\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/a5d13d480cf9626407d8d014295e0e955154251c)
![{\displaystyle {\begin{aligned}\cos {\frac {2\pi }{35}}=&{\frac {1}{6}}\left({\tfrac {1+{\sqrt {5}}-{\sqrt {14\left(5-{\sqrt {5}}\right)}}}{4}}+{\sqrt[{3}]{\tfrac {{-7+49{\sqrt {5}}+15{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}}-3{\sqrt {3}}\left(7+7{\sqrt {5}}+{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)i}{8}}}+{\sqrt[{3}]{\tfrac {{-7+49{\sqrt {5}}+15{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}}+3{\sqrt {3}}\left(7+7{\sqrt {5}}+{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)i}{8}}}\right)\\\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/6b13afa575f6e8963e6d411b6eeb8a3e1b8f3073)
