Regular polygon with integral ratio between apothem and side

Question

Let $n$ be an integer. Find at least one $n$ such that the ratio between tha apothem and the side of a regular polygon with $n$ sides is an integer.

I found this problem while I was casually playing with some math.

I managed to get an expression for the wanted ratio, say $R$, that is $R = \frac{1}{2tan(\pi / n)}$, but here I am stucked.

Answer 1

Here (Corollary 5)

http://www.oberlin.edu/faculty/jcalcut/tanpap.pdf

you can find a proof that $\tan(\frac\pi n)$ is irrational for all $n>4$, so that the only rational value $R=\frac12$ is obtained for $n=4$.