How do Equally Sized Spheres Fit into Space? How much volume do spheres take up when filling a rectangular prism of a shape?
I assume it's somewhere in between $\frac{3}{4}\pi r^3$ and $r^3$, but I don't know where.
This might be better if I broke it into two questions:
First: How many spheres can fit into a given space? Like, packed optimally.
Second: Given a random packing of spheres into space, how much volume does each sphere account for?
I think that's just about as clear as I can make the question, sorry for anything confusing.

You can read a lot about these questions on Wikipedia.
Concerning the random version, there are some links at Density of randomly packing a box. The accepted answer links to a paper that "focuses on spheres".
