Smooth approximation of polyhedral metric!

do someone have an idea, how to prove that :

For every polyhedral metric $$(S,d)$$ of curvature $$\leq -1$$, (S=surface), there exists a sequence of smooth metrics, converging uniformly to $$(S,d)$$.

And why the metric in the neigbourhood of any singularity will be of the shape $$dr^2+f^2(r)d\phi^2$$ with $$f$$ a suitable function.

Thnak you for any answer