Academy of Sciences of Heroesland

You're confusing a sphere with a ball, sir. A body cannot have the same dimension as the surface that bounds it. The bounding surface has one dimension less. Three-dimensional objects have volume, while two-dimensional objects have area. I've seen a formula for the area of a sphere, but not a formula for its volume.

Well, I was just saying that I don't take the dimension of a manifold as the dimension of a figure, so I'm not confusing anything.
A figure is like a submanifold. Of course, the dimension of the submanifold doesn't have to match the dimension of the manifold, but a figure is also distinguished by its order in the space in which it is located. In addition to the two angles we can introduce by introducing coordinates on the sphere, the sphere itself has at least a radius (and, in general, also the coordinates of the center, although this is not so important). The main thing is that the radius determines the properties of the sphere. And that's 3D, no matter how you look at it.
But this is just terminology, of course. As you can already see, it's not just you and me who haven't decided on it