# Regular polygons with rational area

Problem:

Find all natural numbers $$n\ge 3$$ such that the area of a regular $$n$$-gon of radius $$1$$ is rational.

Given a circle of radius $$1$$, the only regular polygons inscribed in it with integer area are the square and the dodecagon. The area of the former is $$2$$ and the later's, $$3$$. Since $$\pi<4$$ there are no more examples. But this bound says nothing about if the area can be rational for other polygons.

The feeling is that there are no more examples with rational area (actually, I learned recently about the dodecagon and I found it quite surprising), but I don't know anything about a proof. I assume that it would involve the known theory about constructible numbers, but I'm not really sure.

## 1 Answer

The area of a regular $$n$$-gon inscribed in a unit circle $$n\sin\left(\frac{2\pi}{n}\right)$$ which is rational iff $$\sin\left(\frac{2\pi}{n}\right)$$ is. But it is well known that $$\sin(q\pi)\in\mathbb{Q}$$ is solvable for rational $$q$$ only when $$q=0,\frac{1}{6},\frac{1}{2}$$ or trivial variations thereof.