# Packing identical cylinders into a rectangular prism

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Ege Arsan on 22 Dec 2021
Commented: Ege Arsan on 23 Dec 2021
Dear all,
I need to pack identical cylinders into a rectangular prism (e. g. figure). Is there a way to do this without identifying the cylinders one by one? Can someone help me with this problem?
Thank you!
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Ege Arsan on 22 Dec 2021
As you said, I am trying to fit a maximum number of fixed-size cylinders in to a fixed volume. I also need to find the used volume by the cylinders therefore the cylinder height is relevant to the problem.
So the constraints are the fixed volume, dimensions of the cylinders, the gaps between the cylinders and the gap to the walls.

Image Analyst on 22 Dec 2021
I gave you a way to pack them in there tightly in your other question. You can easily adapt it if you want spaces between the cylinders. The volume is just the area times the height of course.
Ege Arsan on 23 Dec 2021
There isn't any error now. How should I plot them now?

John D'Errico on 22 Dec 2021
Sorry, but no, there is no simple mathematical formula to perform maximal circle packing of a known radius circle into a given domain. Since your cylinders will be the height of the domain, the height is known. And that means this is just a circle packing problem, in a rectangular domain, so just a 2-d problem. A hexagonal close packed solution is probably at least close to optimal, but along the edges, you may be able to do just a bit better, depending on the dimensions of the rect and the circle radius.
If you then say you don't know the orientation of the cylinders, so you don't know the height, then it is just THREE independent circle packing problems. Solve it three times, once along each axis of the domain. In each case, it becomes a circle packing problem in a rect.
Once you know how many circles can fit into the rect, computing the volume is trivial. Just multiply the area of one circle with the height, then multiply by the number of circles you were able to fit inside. But there is no formula to give you the circle locations or the number of circles.

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