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: enter image description here 2 circles on hexagon

enter image description here 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
  • $\begingroup$ Thanks for the additional info! $\endgroup$ – Ingix Jun 17 '19 at 9:14
  • $\begingroup$ In your first diagram, what prevents the polygon being vastly larger than my monitor, with both circles lying on the same edge, (so that the number of vertices is unknow-able)? $\endgroup$ – Eric Towers Jun 17 '19 at 9:15
  • $\begingroup$ @EricTowers the vertices count is predetermined based on separate data so it cannot be predicted. But its unlikely to have more than 20 vertices. And also there will be an additional step, that if the polygon gets too large, it will be split into smaller inner and larger outer polygon. So unless there are thousands of circles, it will not grow too large. $\endgroup$ – Marko Taht Jun 17 '19 at 10:04
  • $\begingroup$ What does size mean? Edge lenght? I think I read something like that in the body. How can a regular Polygon be "minimal"? The circles' centers being on the perimeter does not suffice as information. Being on the vertices could solve the problem. If you have n non-overlapping congruent circles on the n vertices of a regural n-gon. The edge of the polygon must be at least(or more_ equal in lenght with twice the radius. Exactly equal if they are tangent one on another. We need to know the circles' positions to give you an answer. $\endgroup$ – George Ntoulos Jun 17 '19 at 11:06
  • $\begingroup$ We need to know the number or vertices of the regular polygon, the number of circles, hopefully if the circles are tangent one on another. And the location of their centres.(Vertices probably). Maybe give qualitative information on the circles' radii or say if they are congruent. $\endgroup$ – George Ntoulos Jun 17 '19 at 11:06

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.

  • $\begingroup$ i added the conditions. $\endgroup$ – Marko Taht Jun 17 '19 at 9:14

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