I have been looking for a coffee mug that will hold at least 12 oz of coffee. On ebay, I see things like mug is 3 1/2 inches across and 4 inches deep. This does not tell me a thing about how many ounces this cup will hold. Worse yet I do not know if these are inside or outside numbers, I've learned the hard way you cannot judge anything by the picture. So my question is: Can these figures be of any use in figuring out how many ounces a cup will hold?
Yes, the figures are needed to figure out the volume. It is a standard geometry problem.
First, let's see about the outside vs. inside problem. It is likely that they are telling you the outside measurements, and anyway, in your case it is better to underestimate the volume than to overestimate. So before calculating, you need to make a guess about wall thickness. Now I have seen some large earthenware mugs with thick walls, but for standard ceramics, a quarter of an inch is plenty thick. So you can assume that the inside of the mug is 3 inch across (3.5 - 2*0.25, because you have one wall on each side) and 3.75 inch deep (4-0.25, not 2*0.25 because you only have to put away the bottom) and be on the safe side.
I could derive the formula for you step by step, but I assume you didn't come here for a geometry lesson (if you want to learn the calculation, see https://www.mathsisfun.com/geometry/cylinder.html, and if you are stuck, you can ask questions on math.stackoverflow.com). So my suggestion is to head over to Wolfram alpha and ask. This is a site which tries to answer math problems given in human language. The query
volume of cylinder 3 inches across 3.75 inches high returns 26.51 cubic inches. It has some conversions to other units, but not ounces. So we ask it a second query,
26.51 cubic inches in ounces and it tells us that it is 14.69 fluid ounces.
So yes, it will hold more than your desired 12 ounces. In reality, it will probably hold quite a bit more than the 14.69, because walls are likely thinner than a quarter of an inch. If you are looking at a different mug, plug in the numbers into Wolfram alpha and you will find it out for yourself.