I think you are looking for Lebesque measure, wikipage.
Quote: “For lower dimensions n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n-volume, hypervolume, or simply volume.”
You’d just continue saying ‘volume’, alternatively ‘k-dimensional volume’ or ‘volume of the n-dimensional object’. Like for spheres: https://en.m.wikipedia.org/wiki/Volume_of_an_n-ball
The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is:[1]
I’m going to start calling area “2-dimensional volume”
Only if you also call length “1-dimensional volume”.
What happens if I turn the dimensional volume up to 11?
Well, you could just make 10 higher and make that the highest
