Is there any formula to calculate how many circles of radius r fit in a single bigger circle of radius R?
[Math] How many circles of radius r fit in a single bigger circle of radius R
circlescombinatorial-geometrygeometrypacking-problem
Related Solutions
This is a disk covering problem. As you have stated it, it is not quite as difficult as some others.
The first task is to find the minimum number of small circles which cover the circumference of the bigger circle rather than the whole area. If this is $m$ then it will be impossible to have a regular $m$-gon with edges of length $2r$ fitting strictly inside the circle of radius $2r$. This implies $m \ge 6$.
It turns out to be just possible to cover the circumference of the bigger circle with 6 smaller circles, and ignoring symmetries there is only one way to do it (you get a regular hexagon whose vertices lie on the bigger circle and whose edges are diameters of the smaller circles). But this leaves an uncovered area in the middle, which needs at least one (and in fact exactly one) more.
So the answer is seven smaller circles.
If you connect the centers of two adjacent little circles and the center of the big one, you'll get a triangle. The sides of this triangle have lengths $r+R, r+R$ and $2r$. A little trigonometry will get you that the top angle is
$$\theta=2\arcsin\left(\frac{r}{r+R}\right) \; .$$
Since you want the small circles to form a closed ring around the big circle, this angle should enter an integer amount of times in $360°$ (or $2\pi$ if you work in radians). Thus,
$$\theta=360°/n \; .$$
From this, you can compute that
$$r=\frac{R \sin(180°/n)}{1-\sin(180°/n)} \; .$$
Here's a plot of the result for $n=14$:
Here's the code in Scilab:
n=14;
R=1;
r=R*sin(%pi/n)/(1-sin(%pi/n));
theta=2*%pi*(0:999)/1000;
plot(Rcos(theta),Rsin(theta));
hold on;
for k=0:(n-1),
plot((r+R)*cos(2*k*%pi/n)+r*cos(theta),(r+R)*sin(2*k*%pi/n)+r*sin(theta));
end;
hold off;
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Best Answer
It is a very complicated and investigated problem, usually formulated in the form: given a natural number $n$, which is the smallest radius $R=R(n)$ of the disk containing $n$ non-overlapping unit disks (that is, disks of radius 1). You can see results for small $n$ ($\le 20$) at Packing Center by Erich Friedman. For large $n$ ($\gg 10$) it is practically impossible to obtain tight lower bounds for $R(n)$ and upper bounds are found by computers, trying to find tighter and more tighter packings. I can share with you some articles and a program Pack 1.0 related with this problem. I upload them when my file sharing website will work.