I have the following given values:
- area of the big circle
- values ranging from 1 to 100 (it can be more or less)
I need to create a circle for each value, where its size is proportional to its value. For example, a value of 5 will become a circle that is smaller, while a value of 20 will become a larger circle.
Then, all these circles should fit inside the big circle.
My goal is to find the radius of each circle.
Suppose you have 4 circles of radius $r$ each at the corner of a square having side length =$2r$.
Then the area of this square is $4r^2$. One quarter of each of these 4 circles lies inside the square totaling an area of an entire circle, $\pi r^2$. So for a given square, circles packed in this way will only cover $\frac{\pi}{4}$ of the area.
Repeat this with a regular triangle and we find the corresponding ratio to be $\frac{\pi}{2\sqrt{3}}$
Each of these processes corresponds to a way of packing circles into an area. However you pack the circles, some gaps will be present and with a signature ratio.
So given a square of side length $L$, how many circles of radius $r$ can fit?
The area of the square that can be occupied by Square Corner packing is $\frac{\pi}{4}L^2$
Divide this by the area of the circle to get $N=\frac{L^2}{4r^2}$. So N circles will fit.
The same principle applies for circles. Suppose you have a big circle with radius $r_1$ and you want to fill it with circles of radius $r_2$.
Then $$N=\frac{(\frac{\pi}{4})\pi r_1^2}{\pi r_2^2}=\frac{\pi r_1^2}{4r_2^2}$$
There are different ways of packing, and radii can be non-uniform, but once you know the packing fractions, you can usually sort things out.