# how many items can i fit in my box

I have a cuboid with a width of 500mm, a height of 500mm and a length of 1000mm.

I need to know how many 38mm balls i will need to fill the cuboid.

Any help would be greatly appreciated.

Thank you.

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Three questions: (1) is 38mm the radius? (2) what have you tried so far? (3) is this a homework problem? I also took off the measure-theory tag since that wasn't the right way to classify this problem. –  crf Sep 12 '12 at 10:23
radius is 38mm (its a sphere) the answer i have at the moment is 4394, but i am not sure if this is correct. Sorry i couldn't find a correct tag. –  dan Sep 12 '12 at 10:27

if i am not wrong the total numbers of balls will be "4394"

  26 balls length wise


{ 1000/38= 26(approx) }

13 balls width wise { 500/38= 13(approx) }

and 13 height wise { 500/38= 13(approx) } which will lead you to 26*13*13 = 4394 balls (approx)

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This assumes square packing, which is easily shown to be sub-optimal, at least for infinite packing. How small the grid has to be for this to take effect, I do not know. –  Arthur Sep 12 '12 at 11:17
thats the answer i had, thank you for clarifying! –  dan Sep 12 '12 at 11:20
@danyo no Problem :) –  Ali Sep 12 '12 at 12:55

Wikipedia says if you just pour the spheres into the box you will achieve a density of $0.609$ to $0.625$, leading to $5299$ to $5438$. If you shake the box it will be $0.625$ to $0.641$, or $5438$ to $5577$ balls. If you pack them perfectly (and ignoring the edge effects) you get $0.7401$ for $6439$ balls. All are much better than cubic packing.

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