Find the regular $n$ -side polygon of A Constant Area that can contain Most Number of Circles I have a constant area $A$   , and I can mold that area into a regular $n$ -side polygon, where $n\geq 3$. The issue now is how to find the $n$  such that it can contain the most number of circles, each with a constant radius $R$ ?
Edit to clarify the question: the area must be a regular $n$ -side polygon. Sorry for not making this clear
 A: If $n$ is unconstrained, as it seems to be in the question, then the polygon can be taken to be an $\epsilon$-accurate approximation to the optimal figure, and we can ask directly what that figure (or figures) look like.

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*If you don't assume the polygon is convex, the answer is trivial, in that you can get as close as desired to a collection of $k$ disjoint circles connected by very thin tubes. In this version of the problem the only constraint is that $A > k \pi R^2$ from which the maximum $k$ is easily determined.


*If convexity is required, finding the tightest configuration of circles -- the one whose convex hull is a minimum-area figure containing $k$ or more discs of given size -- is a hard nonlinear optimization problem which is a variant of the classical problem of circle packing (finding the smallest circle enclosing $k$ unit discs).   Recent experiments indicate that the optima, even for large $k$, do not have a connectivity pattern approximating the optimal lattice packing.


*If the polygon is convex and you are satisfied with an asymptotically optimal solution, start from a hexagonal packing of circles of radius $R$ in the whole plane, and draw a circle of area $A$ that encloses as many of these as possible, then take the convex hull of the circles inside, then approximate the convex hull closely enough by a polygon.
(added: for asymmetry in high density finite packings up to n=348, see https://arxiv.org/abs/1002.0604 and a long series of theory papers by the same authors.  Best known packings of small numbers of disks in circles, hexagons, squares, and other shapes are displayed at: https://erich-friedman.github.io/packing/index.html  .)
