Is there a geometric relationship between a solids surface area and its volume?
For a specific kind of solid, yes. In general, no. For example, a cube with a side length of s will have a volume of s^3 and a surface area of 6s^2. So, for any cube, the volume and surface area will be in the ratio of s/6. For a right cylinder, the volume is (pi)r^2*h, while the surface area is given by 2(pi)r^2 + 2(pi)r*h, which through some algebraic manipulation can be reduced to 2(r+h)(pi*r). So the ratio of volume to surface area for a cylinder will be r*h to 2(r+h)… For other solids, you could observe similar relationships, but in general, the relationship depends on the shape of the solid, so you could not make a definitive statement about the surface area enclosing a certain volume unless you also knew what shape that volume was in.
