I start by stating the problem, which is already hinted in the title of the question. I do believe it is a research-level question.

> Let us fix a regular $n$-gon with area $1$. What is the smallest possible area of a regular $m$-gon, so that it can cover the $n$-gon.

Of course, *cover* means *to contain as a set*. Of course, we can ignore the ill-defined scenarios, and fix $n\geq 3$ and $m\geq 3$. There is a number of situations on which it is possible to give a explicit formula (using sines and cosines) in terms of $n$ and $m$ of the desired smallest area. 

Although this problem has a quite simple statement, it is not trivial, for example what the answer is for $n=5$ and $m=7$ or for $n=7$ and $m=5$. Notice that the question only asks for a covering, so that it is not required to the polygons to be concentric.

When $m=4$ I could get an answer for all even $n$, and I think I can also solve it for odd $n$. Also, the case on which $n\mid m$ is, I think, much easier since the intuitive configuration does work, and a formula can be given.

Any thoughts on this? Is there any hope to give a formula using sums and products of sines and cosines of ugly rational multiples of $\pi$ for arbitrary $n$ and $m$?