Calculate the size of a polygon given the number and radius of circles that are on the perimeter

Question

How can i calculate the size of a regular polygon if i have N circles of same size and all the circles must be on the perimeter of the polygon.

Examples: 2 circles on hexagon

3 circles on hexagon

Is there a equation that given the number of vertices on polygon (>= 3) and the number of circles (>=2) and radius of the circle, will output me the edge length of a such a regular polygon? Also it would be bonus if there would be a way to find the positions of the circles also.

Conditions for the hexagon and circle placement:

1. Hexagon must be minimal
2. Circles must not overlap
3. Circle centers must be on perimeter

Answer 1

There is not enough information to calculate an exact length, because there is lot's of "wiggle room"! In the first picuture, for example, slightly increasing or decreasing the polygon side length still seems to produce an "acceptable" configuration.

Of course, what your problem description is missing is a rigorous definition of what "acceptable" is. From the pictures I assume

1. The centers of the circles must be on the perimeter of the polygon, and
2. the circles cannot overlap.

Using these conditions alone means that any sufficiently large edge length will work (I see no problem putting 100 circles of radius 1cm onto a regular triangle with side lengh 100km, for example).

So please specify the exact conditions for "acceptable" configurations, and if it isn't more than what I wrote, I think the only reasonable question would be to ask for the smallest edge length that can fulfill those conditions.