  Let X be a closed oriented Riemann surface of genus > 1 of constant negative
curvature -1. A surface containing a disk of maximal radius is an optimal
surface. This paper gives exact formulae for the number of optimal surfaces of
genus > 3 up to orientation-preserving isometry. We show that the automorphism
group of such a surface is always cyclic of order 1,2,3 or 6. We also describe
a combinatorial structure of nonorientable hyperbolic optimal surfaces.
