# How many circles of radius r fit in a single bigger circle of radius R?

Is there any formula to calculate how many circles of radius r fit in a single bigger circle of radius R?

• circle packing in a circle or in general circle packing is known to be a hard problem. Only a few solutions is known. You exercise must be asking something more specific or it will be impossible for high school to solve. – achille hui Sep 15 '13 at 2:53
• And with advanced math? Is it possible? – Lucas Cleto Sep 15 '13 at 21:55
• As I have said, only few solutions are known. Even with advanced math, no one in the world know the full answer for arbitrary $r$ and $R$. What is the exact statement of your exercise? Does it just ask you for a qualitative description of the tightest arrangement? or to give an estimate/a lower bound/an upper bound for the numbers you can fit in the bigger circle? or simply something else. – achille hui Sep 16 '13 at 0:20
• There is the obvious upper bound of $\left(\frac{R}{r}\right)^2$ but even rounding that down to the nearest integer will rarely give you the correct maximum number. – fretty Sep 16 '13 at 11:54

It is a very complicated and investigated problem, usually formulated in the form: given a natural number $n$, which is the smallest radius $R=R(n)$ of the disk containing $n$ non-overlapping unit disks (that is, disks of radius 1). You can see results for small $n$ ($\le 20$) at Packing Center by Erich Friedman. For large $n$ ($\gg 10$) it is practically impossible to obtain tight lower bounds for $R(n)$ and upper bounds are found by computers, trying to find tighter and more tighter packings. I can share with you some articles and a program Pack 1.0 related with this problem. I upload them when my file sharing website will work.