# Regular polygon with integral ratio between apothem and side

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.

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 $$n=4$$.