Okay, what 2D shapes is a cone made of?

So I just read that a sphere is a 2D shape (in mathematics/geometry, at least) and is actually the surface of a ball (which is the 3D shape). It got me wondering, what is a cone made of ... a circle as the base, but what is the other part called (unless that *is* a cone and the 3D shape I'm thinking of is not a cone but has some other name)?