Conformal mappings from $U$ to $V$, both subsets of $\mathbb{C}$, locally preserve angles. But, in general, such mappings neither preserve areas nor preserve perimeters.

. Are there examples of analytic conformal mappings that preserve areas but not perimeters? Or vice versa?Q

^{ The conformal mapping $w=z^2$ in rectangular coordinates: (John Mathews.) }