Say that a square $S$ is said to be inscribed in a regular polygon $P$ if all the four vertices of $S$ lie on the boundary of $P$. It is well-known that one can inscribe a square in a regular $n$-gon for $n\geq 5$.
I would like to know, up to rotational symmetry, how many distinct squares can be inscribed? For example, in a hexagon only one square can be inscribed.
Second question: What is the ratio of their side lengths?
Best Answer
If $n$ is a multiple of $4$, then every couple of opposite points (with respect to the center) on the polygon can be taken as endpoints of a diagonal of an inscribed square, so in this case we have infinitely many solutions.
In the other cases, it is not difficult to prove that a solution is possible, where the inscribed square has a side parallel to a side of the polygon. I think there are no other solutions, because to find the inscribed square one has to solve a system of linear equations, which can be indeterminate (as in the case when $n$ is a multiple of $4$) but otherwise cannot have more than one solution.