Is there a list of 2-dimensional hyperbolic orbifolds obtained from reflection groups (such as the double of a hyperbolic triangle with angles $\pi/p$, etc.) of small area, for instance area smaller than $\pi/2$?



hyperbolic orbifolds of small area
Yes. One can extend the Gauss-Bonnet theorem to 2-orbifolds since but area an orbifold Euler characteristic behave well under the covering map.


To find all hyperbolic 2-orbifolds with area smaller than $\pi/2$, we need to get a list of orbifolds $\{O\_i\}$ $0>\chi(O\_i)>-1/4$.


Thankfully, this is not a horrible task. Background and pictures for this computation is in [Chapter 13 of Thurston's notes](http://library.msri.org/books/gt3m/PDF/13.pdf), although there are certainly older sources. To set notation, $O$ is our orbifold, $X\_O$ is the underlying topological space of this orbifold. Also, $O$ can have three types of singularities cone points of order $n\_j$ (points fixed locally by rotations of order $n\_j$ and not any reflections), points fixed by reflections but not any rotations, and corner reflections (points fixed locally by a dihedral group). 
Note, fixed by reflections but not fixed by a dihedral group will not contribute to this computation and if a point at infinity is fixed at least to reflections we say the dihedral group that fixes it is infinite.
If an orbifold $O$ has $n$ cone points each fixed by a cyclic group of order $r\_i$ and $m$ corner points each fixed by a dihedral group
of order $t\_j$ then the orbifold Euler characteristic is:


$$\chi(O)= \chi(X\_O) - \Sigma\_{i=1} ^n (1-1/r\_i) -\Sigma\_{j=1} ^m (1/2-1/t\_j) $$


(if $t\_j$ is $\infty$ of course $1/t\_j$ should be treated as $0$).


Now in order to get a orbifold $O$ such that $0>\chi(O\_i)>-1/4$. The underlying space has to be $S^2,D^2,RP^2$.


Also, adapting the same notation as the reference say $Q(r\_1,...,r\_n;t\_1,..,t\_m)$ is the orbifold with base space $Q$ cone points of orders $r\_i$ and corner points of orders $t\_j$.


If there are corner points, then the underlying space $X\_O$ must be $D^2$. 
Each corner point contributes $-(t\_j-2)/(2t\_j)\leq -1/4$ to the Euler characteristic, and so there are at most 4 of them. 


Each cone point contributes $-(r\_i-1)/r\_i\leq -1/2$ to Euler characteristic. So there are at most 2 cone points if the underlying space is $D^2$ or $RP^2$ and at most 4 if the base space is $S^2$. 


From that, one should be able to make a list. 


Note: If one allows equality, $\chi(O)=-1/4$, then an orbifold with underlying space an annulus or Mobius band and one corner point of order 4 should also be included on the list. As well as the orbifold with underlying space a disk and 5 corner points of order 4. There are few others as well, but they all have underlying space $S^2,D^2$ or $RP^2$.


