In a $1$ dimensional object, the name for the region enclosed by it is the length of the object. In a $2$ dimensional object, the name for the region is the area of the object. In a $3$ dimensional object, the name is the volume of the object. What is the name in a $4$ dimensional object? Hypervolume? In general, what can we call the name of this region in a $n$ dimensional object? (“the region enclosed by this $n$ dimensional object” seems too long and wordy).
I would say that just volume is actually a fine choice for any dimension $\geq 3$, but if you really want to emphasize the relevant dimension, you could say $n$-volume or $n$-hypervolume. Take a look at the Wikipedia page on Lebesgue measure.
I think good catch-all terms (i.e., for dimensions $1$ and $2$ as well) would be measure or content.
How about $n$-volume? Many objects carry the name from 3 dimensions as they are generalized to higher dimensions: $n$-cube, $n$-ball whose boundary is an $(n – 1)$-sphere come to mind.