Surface integral - How to parameterize efficiently

Question

I have a question regarding parameterization of surface integrals. How do I know which parameter I input?
Lets say I have the Tetrahedron with:
$x+y+z=1$
bounded by $z=0$, $y=0$, $x=0$.
My teacher did this parameterization:
$x=u$
$y=v$
$z=0$.
How do I know when to do parameterization like that:
$x=ucos(v)$
$y=usin(v)$
z=something.
Or something like the $x=u$, $y=v$ and such..
Any tip will be welcomed.