I need some math help: How can I calculate circle packing within a circle?

Question

I need some help:
What I'm dealing with are tubes with a 3 cm diameter. I need a way to calculate how efficient the placement of these tubes within certain diameter circular areas would be.

I think the best way would be for all the tubes to be tangent to their 'neighbors' and thus create a honeycomb-hexagon shape within the circular area.

As examples (there are several more calculations that I need to do), I need to find the number of the 3 cm tubes that I can fit together within a 20 cm-diameter circular area. Also, I'd like to do sort of a backwards calculation from that and find how much circular area 30 of the tubes would occupy. As I've said, there are plenty more to do, but if someone would explain to me a procedure for working the above calculations, I can do the rest on my own.

Finally, I thought it might be helpful to visualize, so I looked for circle packing graph paper, but I couldnt find any. If you find a link to that it would be very helpful.

THANKS!

Answer 1

You are on the right track.
This a geometrical problem that requires drawings.
Check this link http://departments.kings.edu/chemlab/chemlab_v2/packgeo.html

I'll get back to you later

so later it is

3--------R=r(1+2/3^.5)
7--------R=3r
19------R=5r
37------R=7r
61------R=9r

Answer 2

This is a common problem in solid state physics and most textbooks answer it. Forget for the moment the diameter of each circle. Take the radius of the large circle to be rl (large) and the radius of smaller circles to be rs (small). Now you know the area of the large circle, pi*rl^2. Now maximize the number of smaller circles and then plug in rs value.