How would I describe a circle in the same way as we formulated the lateral area of a cylinder (below)? My first guess would be $\phi=(R\cos u,R\sin u,k)'$ , ($k$ being the $z$-plane where the circle would be) but then I cannot plot it because there is no '$v$'.
Edit: I would be plotting it in Matlab using ezsurf()
I think you are trying to graph a disk, not a circle. You may parametrize it as $\phi : [0, 2\pi]\times [0,R] \to \Bbb R^3,\quad \phi(u,v) = (v \cos(u), v\sin(u), k)$