How many circles of diameter d on average in area of size A?

Hi,

I have an unlimited large plane. I am trying to calculated how many circles with a certain diameter d can fit into any square of size SxS on the plane. I think this should be the maximum density. So the question is NOT "How many circles can I fit into square of size x?", as it does not matter if the circles overlap the boundaries of the square.

(The setting is: How do I calculate how many people that try to keep distance d between their cores (modelled as circles) can fit into an area of size 1: What is the maximum density?)

I have looked into circle packing but that seems to calculate slightly different things.

I think it's either

2 / sqrt(3 * d * d)

or

2 / (sqrt(3) * d * d).

Any help (even a search word for google!) would be appreciated!

Re: How many circles of diameter d on average in area of size A?

The densest packing is indeed the circle packing problem, and the ratio of circles to total area is

Both of your two alternative choices have a problem with dimensions. The first has units of 1/length, and the second 1/length^2, whereas the answer should be dimensionless.

1 Attachment(s)

Re: How many circles of diameter d on average in area of size A?

I don't understand?

The question is : How many circles with diameter d can fit into an area of 1 x 1.The answer should always contain d?

Attachment 26796

In this situation, how many circles (so that could be 1 + 1 + 1 ... + .5 + .3 + .7) with diameter d fit into the red square of size 1 x 1?