Given $x$ the number of circles of radius $r$, what is a good approximate size of the radius of a bigger circle which they fit in?
To explain in actual problem terms. I want to move units in a video games which take up a certain amount of area around, I don't want those units overlapping so I want the points there told to move to to be spaced far enough apart to make sure they aren't. I know how to make sure each point is far enough apart but i'm picking points randomly in a circle and then checking to make sure they are far enough apart.
The problem is if a circle is too small in size there simply isn't enough valid movement points for all the units and you get an infinite loop endlessly trying to find an answer that doesn't exist. If the circle is too large you end up with units spaced out over vast areas way out of proportion.
I do not need a perfect minimum radius answer I just need to be certain the amount will fit while not being too large. To show that I did think about it my rough guess was.
the square root of (amount needed) * (radius of each circle * 2)