# How many circles of a given radius can be packed into a given rectangular box?

I've just came back from my Mathematics of Packing and Shipping lecture, and I've run into a problem I've been trying to figure out.

Let's say I have a rectangle of length $l$ and width $w$.

Is there a simple equation that can be used to show me how many circles of radius $r$ can be packed into the rectangle, in the optimal way? So that no circles overlap. ($r$ is less than both $l$ and $w$)

I'm rather in the dark as to what the optimum method of packing circles together in the least amount of space is, for a given shape.

An equation with a non-integer output is useful to me as long as the truncated (rounded down) value is the true answer.

(I'm not that interested in how the circles would be packed, as I am going to go into business and only want to know how much I can demand from the packers I hire to pack my product)

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I had an answer before, but I looked into it a bit more and my answer was incorrect so I removed it. This link may be of help: en.wikipedia.org/wiki/Circle_packing_in_a_square –  Cam Jul 26 '10 at 6:30
@Cam: Looks like there's no optimal solution yet. Maybe you could just put this comment as an answer. –  KennyTM Jul 26 '10 at 6:34
Might be a good question to work out how to answer problems which actually aren't solved yet in advanced maths. (if there is not an optimal solution yet) –  Justin L. Jul 26 '10 at 7:01
@KennyTM: Sure. –  Cam Jul 26 '10 at 12:31