The space that is enclosed by a unit sphere in dimension 1 is the interval (-1, 1). In physics this is measured in meters. This is referred to as length. The space enclosed by the unit sphere in dimension 2 is measured in meters squared. This is referred to as area. In dimension 3 we call it volume and measure it in meters cubed.
In dimensions 4 through infinity we quickly come to a problem. Do we come up with unique names to refer to amount of space enclosed by a unit sphere? Mathematicians have decided to just use the word volume. We rely on context to make it clear what the dimension is. Similarly we use the word n-sphere to refer to the collection of points in n-dimensional space that are exactly 1 unit from the origin.
In my applied mathematics education the highest-dimensional unit of measurement of a given dimensionality was referred to generally as ‘content’: length is the content of 1D objects, area is the content of 2D objects, volume is the content of 3D objects, hypervolume is the content of 4D objects, and henceforth you just start numbering them...
In dimensions 4 through infinity we quickly come to a problem. Do we come up with unique names to refer to amount of space enclosed by a unit sphere? Mathematicians have decided to just use the word volume. We rely on context to make it clear what the dimension is. Similarly we use the word n-sphere to refer to the collection of points in n-dimensional space that are exactly 1 unit from the origin.
Surface area of an n-sphere is used and studied.