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I have been give a particular task with packing hexagonal shapes with radius 0.105m, into different circular areas. This is not a 3D problem, and I have been trying to search for answers on the topic of "packing" but haven't seemed to find any that fit my requirements.

So the idea is I need the shapes to stay flat and they cannot be manipulated. I want to maximize the area covered in the circle with a given amount of hexagonal shapes. For instance in a 2 foot=0.6096m diameter circular shape I can fit around 7 hexagonal shapes. Even though the logical answer is around 21 the geometry behind it allows me to comfortably pack 7.

The same sort of experiment took place with a 4.64m diamater circlular shape. Mathematically from formulas I can fit 539 of these shapes comfortably. (Area hexagon =0.0286m^2 and Area of circle=16.97m^2, but 16.97/0.0286=593?! So a difference of 44 shapes not needed cause they dont fit.

I guess my overall question is their a relationship between this particular hexagons area and different areas of any circles? I don't want one hexagon inscribed in one circle, it's many little hexagons in any particular sized circle.

Thanks in advance for any help,

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# Geometry Problem involving packing Hexagons into Circles

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