What symmetry will you find in a surface that has an equation of the form $\displaystyle \rho=f(\phi)$ in spherical coordinates? Please explain.
Answer: RADIAL symmetry. Notice that $\displaystyle \rho$ represents the "radius" at that point on the 3D figure, and $\displaystyle \phi$ represents the angle off the "north pole" and nowhere in the function is $\displaystyle \theta$ mentioned. Therefore, the function will have the same value regardless of $\displaystyle \theta$, so you can rotate it about the z-axis without changing it's shape, therefore, you have radial symmetry.