Max Squares Fitting in a Circle: Proof & Formulas

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SUMMARY

The maximum number of squares that can fit inside a circle is determined by the radius (X) of the circle and the side length (Y) of the squares. While no general formula exists for all cases, discussions suggest that gapless packing may provide bounds for the number of squares, particularly in large circles. The relationship between the radius and the side length influences the packing efficiency, with larger circles potentially allowing for optimal packing configurations. Resources such as the packing problem links provided offer further insights into this mathematical challenge.

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  • Understanding of geometric packing problems
  • Familiarity with mathematical proofs and formulas
  • Basic knowledge of circle and square properties
  • Ability to interpret mathematical resources and literature
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  • Explore "gapless packing strategies" for optimal configurations
  • Study "mathematical proofs in geometry" for deeper understanding
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simpleton
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Hi,

Given that the radius of a circle is X, and the side length of a square is Y, what is the maximum number of squares you can fill inside this circle, provided that the squares do not overlap? If you know of a general formula or something, can you please tell me the proof or give me a link to the proof or something?

Thanks a lot.
 
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Have a look at
http://www2.stetson.edu/~efriedma/packing.html
http://en.wikipedia.org/wiki/Packing_problem
http://mathworld.wolfram.com/SquarePacking.html
which gives you the answer. Consider scaling to connect it to your problem with X and Y.

I haven't heard of a general formula. But with large circles one could write down an (ugly?) formula for the bounds to the number of squares by assuming gapless packing.

Maybe for very large circles the gapless packing is even optimal? Not sure about how much the boundaries matter.
 
Gerenuk said:
Maybe for very large circles the gapless packing is even optimal? Not sure about how much the boundaries matter.

Probably "square root"-ly. So not much if you're concerned about the percentage filled, but a lot if you care about the amount not filled.
 

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