This is something that's always bothered me. I am well aware that you can easily see why this is the case with math. I mean, even in the 2-D case, take a square with side length $1$, and it has a perimeter of $4$ and an area of $1$. Now take a rectangle with side lengths $0.5$ and $1.5$, then the perimeter is $4$, but now the area is $0.75$.
But where did all the extra area go? We have the same amount of material there. Intuitively I'd like to say that the same area is covered, but it's just distributed differently, but this clearly isn't the case. I know that different shapes with the same surface area have different volumes, but I can't picture why. I can take a sheet of (ideal) paper, turn it to a sphere, and then turn it into a cube, and they'll have different volumes, despite using the same sheet of paper.
To extend on this question a little further, what makes one shape "better" with volume than another?
I hope this isn't a remarkably trivial issue wherein I'm missing something obvious.