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I want to find for what $n\in \mathbb{N}$ a $n$-sided polygon has rational area, assuming the polygons' "long" radius is $1$. This reduces to whether or not $\sin\left(\frac{2\pi }{n}\right)$ is rational.

Solutions for $n$ found so far include $1, 2, 4, 12$. Have not found a corresponding sequence on OEIS.

Thanks.

lioness99a
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Ola
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    http://mathworld.wolfram.com/NivensTheorem.html (a link I obtained from http://math.stackexchange.com/questions/87756/when-is-sinx-rational) tl;dr version: you have actually found all of them. – Ian Mar 27 '17 at 12:13
  • @Ian Thanks. So the only regular polygons with rational area are the 4- and 12-gon. If you post an answer i will mark the question answered. – Ola Mar 27 '17 at 12:18

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Because of Niven's theorem (see http://mathworld.wolfram.com/NivensTheorem.html), you have found all such numbers, so the only regular polygons with rational side length and rational area are the 4-gon and the 12-gon.

Ian
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