A detailed write up of what you are looking for is also available in Chapter 13 of Thurston's notes. Specifically, formula 13.3.4 on page 331, has the desired formula
chi(O)=chi(XO)−frac12sumNi=1(1−frac1ni)−sumMj=1(1−frac1mj),chi(O)=chi(XO)−frac12sumNi=1(1−frac1ni)−sumMj=1(1−frac1mj),
where OO is the orbifold, XOXO is it's underlying space, there are N points fixed locally by dihedral groups of orders 2n1,...,2nN2n1,...,2nN and MM points fixed locally by rotations of orders m1,...,mMm1,...,mM.
However, I believe the question addressed (at least in part) by Hurwitz in the proof of Hurwitz's theorem, since the bound the order of an automorphism group H acting on surface of genus g is 168(g − 1) if H includes orientation reversing symmetries and 84(g-1) in the orientable case. The key observation is that the Gauss-Bonnet extends to orbifolds as well.
Computing the Euler characteristic for the quotient of mathbbH2mathbbH2 by the (full) (2,3,7) triangle group is a good way to see this in action.
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