Rational values for $\sin\left(\frac{2\pi }{n}\right)$

Question

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.

Answer 1

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.