How do Equally Sized Spheres Fit into Space?

Question

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.

Answer 1

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".

Answer 2

There are some links here (Rusin's known-math site). Also here (Eppstein's Geometry Junkyard).