Physical representation of volume to surface area I was looking at this XKCD what-if question (the gas mileage part), and started to wonder about the concept of unit cancellation. If we have a shape and try to figure out the ratio between the volume and the surface area, the result is a length. For example, a sphere of radius 10cm has the volume of $\approx 4118 cm^3$ and an area of $\approx 1256 cm^2$. Therefore, the volume : surface area is $\approx 3.3 cm$.
My question is: what is the physical representation of length in this ratio?
 A: The volume of a sphere is $V = \frac{4}{3} \pi r^3$ and the surface area is $A = 4 \pi r^2$. Thus, their ratio is given by: 
$$\frac{V}{A} = \frac{r}{3}$$.
Thus, 3.3 cm is one-third of the radius of the sphere.
A: In this particular case, since $V = \frac43 \pi R^3$ and $A=4\pi R^2$, the ratio $\frac{V}{A}=\frac{R}{3}$.
In general, dimensional analysis (unit matching) is a tricky beast to deal with. It is usually only a valid approach to a problem if you are certain (or nearly certain) that you have identified all the variables which can possibly affect the problem being "solved". It should also be noted that any equations outputted are only valid up to numerical constants, which are hard to determine. The wikipedia page has a reasonably good (though quite dense) description of the ideas behind it:
http://en.wikipedia.org/wiki/Dimensional_analysis#A_simple_example:_period_of_a_harmonic_oscillator
A good example, commonly used, of the process working well is when it is applied to the simple pendulum. In general though, it can produce some unusual results and can't always be trusted. Since this question was inspired by xkcd, I think it's fair to link to this for an example:
http://xkcd.com/687/
