Solving Annoying Circle: Find r When n Circles Don't Overlap

[fisico]

Hi, this is the question:

If the area of a circle C is A and there are n circles of radius r that do not overlap, inscribed on C, what is r?

I was thinking of A = n(pi)r^2, or the area of C equals the sum of the areas of the circles with radius r to get r, but since there is space in between the circles that is not occupied by them, (since the circles do not overlap with each other) then that equation must be wrong.

Help please?

Thank you

 

[acm]

I got r = sqrt(n)/2. I am probably wrong though.

 

[fisico]

How did you get that?

 

[acm]

Let the radius of the large circle be R
Let the radius of the small circle be r

Area of the Large circle (A) = (pi)R^2
Area of the small circles (A_1) = n(pi)r^2

A - A_1 = Gaps between the circles
A - A_1 -> 0 as n -> Infinity

Hence A - A_1 = pi( R^2 - nr^2)
0 = pi( R^2 - nr^2)
R^2 = nr^2
R^2/ n = r^2
R / Sqrt (n) = r

Radius of the large circle = n/2
:. Sqrt (n) /2 = r

 

[fisico]

thanks, but who said anything about n -> infinity? When I said n I meant a number. Remember also that R could be >> r. Do not assume anything I haven't told you except maybe that all adjacent circles are touching themselves without overlaping.

Thanks for the interest though.

 

[0rthodontist]

How do you inscribe a circle on a circle? Isn't the only possible circle inscribed within a circle, the circle itself?

 

[fisico]

Could you imagine 2 circles inside a much bigger circle?

 

[radou]

The area A of the disk C cannot equal any sum of areas of disks of radius r 'inside' C, given the condition that, for every point inside the disk C, there is a disk of radius r to which the given point belongs. One would think this could only be for r -> 0 and n -> infinity, but a circle is a circle, so, seems like this makes no sense, if I'm right.

 

[fisico]

What you're saying I already mentioned explicitely. Who said that every point in the circle needs to be occupied by a circle? I only said that the circles need to be touching without overlapping. The question never even mentioned that every point needs to be occupied. And by the way, I never said r = dr. All quantities except for C given in the problem are numbers.

 

[radou]

There is no such r. Imagine three disks 'inside' the 'big' disk such that every disk shares one point with the other two, i.e. the disks are 'touching'. Then, in order to 'fill' up a part of the area between the three disks with another disk, the other disk has to have a radius r' < r.

 

[StatusX]

From http://en.wikipedia.org/wiki/Packing_problem" :

Some of the more non-trivial circle packing problems are packing unit circles into the smallest possible larger circle. Results up to 20 unit circles have been proved (the first three are considered trivial)[2]. There are many other problems involving packing circles into a particular shape of the smalllest possible size.

http://www.stetson.edu/~efriedma/cirincir/" a source that shows these first 20 packings, but I imagine it would be pretty hard to prove they're optimal. Is this what you meant?

 

[fisico]

Thanks a lot StatusX. That's pretty much it. I hope now Rodou understands that I never said that the entire area has to filled with circles. I only wanted to know the max amount of circles that can be incorporated into a bigger circle. Since form the website there is no clear way to go about it, since it is extremely hard to find and prove an optimal configuration of the position of the circles to maximize number of circles that can be inscribed. (I imagine that when the numbers go really up, it would take the best computers possibly billions of years to find and prove optimal configurations to maximize number of circles inscribed.) That's a really interesting topic that I had no idea about. Thanks a lot in general.