How does volt effect battery size?

Answer 1

In a roundabout way.

Battery in electrics actually means a collection (=two or more) of cells.

Assuming it's new or fully charged, each cell will give a voltage acccording to which chemistry it uses(Ni-Cd, NiMh, Li-Po, etc) and a capacity according to its size.

So the battery size you get is a combination of how many cells you need to reach the voltage that you want, and the capacity that you want.

You could get a whopping voltage out of a battery about the size of a pen if you put together a stack of pea-sized hearing aid batteries, but the capacity wouldn't be much to brag about. In the opposite fashion you could have one cell the size of a soda can that'd have a huge capacity, but not much voltage.

Old lead-acid batteries as in submarines could have cells basically the size of a suitcase. Huge capacity, but only two volts each.