N-gon's circumcircle

Question

In a regular n-gon, the radius of the circumcircle is equal in length to the shortest diagonal. Find the total number of values of n<60 which this can happen.

The answer is n = 12 (only one value), which is convincing. However, could someone give a foolproof method for this problem? Is there any way to show that this solution is THE ONLY solution? Thanks!

Answer 1

We end up with the following figure.

Answer 2

Hint: the shortest diagonal is one that skips over just one vertex. The central angle it subtends is $\alpha = 2 \cdot 2 \pi / n\,$, and its length is $2 R \sin \alpha/2\,$.