We're using volume the whole time. Area is just the two-dimensional equivalent of volume. The two-dimensional equivalent of surface area would be perimeter, which is relatively rarely used.
The more general term for ’area’ and ’volume’ in higher dimensions is ’content’: area is the content of a 2D item, volume is the content of a 3D item, hypervolume is the content of a 4D item, et cetera.