If a 1D figure has length, a 2D figure has area, and a 3D figure has volume, are there names for what's inside 4D, 5D figures and so on?

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."