Projecting from volume to surface area of a sphere Does it make sense to, and is there a formula for the sphere whose surface consists of all the points inside another sphere?
I realize that I'm trying to compare cubed units with squared units here but I think the assumption I'm making is that these spheres are more like clumps of sand, so another way to ask this question is: given a ball of sand, how big is the spherical shell I could construct out of grains of sand?
Hopefully this isn't nonsense.. I have a slight fear that it could be.

The region between two spheres of radius $R_{1} < R_{2}$ has volume $\frac{4}{3}\pi (R_{2}^{3} - R_{1}^{3})$. If it were filled with sand grains of side length $r \ll R_{1}$, and if these grains were rearranged into a spherical shell of radius $R$ and thickness $r$, the shell would have volume (roughly, to within a factor of $2$ or so due to gaps between grains and other "local" irregularities) $4\pi R^{2}r \approx \frac{4}{3}\pi (R_{2}^{3} - R_{1}^{3})$, or radius
$$
R \approx \sqrt{\frac{R_{2}^{3} - R_{1}^{3}}{3r}}.
$$
For instance, a ball $1$ meter in radius ($R_{1} = 0$, $R_{2} = 1$) filled with $1$ millimeter sand grains ($r = 0.001$) would make a spherical shell about
$$
\sqrt{\frac{1}{0.003}} \approx 18.26\ \text{meters}
$$
in radius.
