To visualize what I mean, Let's say we take a bangle of circumference=$2 \pi r$. Let's keep this on the floor, so that the bangle is along a plane perpendicular to the floor. Let's rotate the bangle. Won't the 3d image formed, look like a sphere?
If we consider the amount of angle it has covered, it will be 2$\pi$ (we can only consider half of this, i.e, $\pi$ for the area of the sphere, since it covers the whole area, twice.) So the length along which the bangle moves, would be $\pi$r, to form a sphere.
Hence, I'm getting the surface area, yo be= $2 \pi r*\pi r$ =$2 \pi^2r^2$
Edit: I have added 2 pictures to explain, what I mean
