Imagine some ring-board with a bullseye of radius `r`.  You have some amount of paint `p`, where `1p` can cover `1*pi^2` area of the board (e.g. to paint a square with side length `2`, `p` would have to be at least `4`).  You want to find out how many rings `c` you can paint on the dartboard.
 
To obtain the radius of the circle enclosed by ring `c`, we need to take into account the inner radius.  Blah blah blah math.  The area of an annulus (in terms of pi) is `R^2 - r^2`.  Blah blah blah more math.  Thus, we can determine the area of ring `c`.  Using differential techniques (I hope you were awake in either statistics or calculus), we can determine that the difference between the areas of ring `c` and `c+1` is always `4`.  Thus, to get the amount of `p` needed for `c` rings (also requiring knowledge of `r`), `p(c,r) = 2c^2 + (2r-1)c`.  This is great, except it's backwards; we need to know the amount of rings `c` that can be made with `p`.  So, solve for the `c`!  Here it comes...
 
`c(p,r) = (1/4) * ( sqrt(8p + 4r^2 - 4r + 1) - 2r + 1 )`
 
This equation may yield fractions of circles; we know you can't have part of a circle, so it must be rounded down to the nearest whole circle.