I am reading some lecture notes on Fuschian groups, and one theorem given is that if $\Gamma$ is a group of isometries of the hyperbolic plane, and if $F_1$ and $F_2$ are fundamental domains of $\Gamma$ so that $\mu(F_1) < \infty$ and $\mu(\partial F_1) = \mu(\partial F_2) = 0$, then $\mu(F_1) = \mu(F_2)$.
It's a simple proof by set manipulation, but a key step is that $\mu(F_2^\circ) = \mu(F_2)$, so I was wondering if someone more versed than I could provide a counterexample where $\mu(\partial F_2) > 0$. My investigations are not proving fruitful.