  We study hyperbolic polyhedral surfaces with faces isometric to regular
hyperbolic polygons satisfying that the total angles at vertices are at least
$2\pi.$ The combinatorial information of these surfaces is shown to be
identified with that of Euclidean polyhedral surfaces with negative
combinatorial curvature everywhere. We prove that there is a gap between areas
of non-smooth hyperbolic polyhedral surfaces and the area of smooth hyperbolic
surfaces. The numerical result for the gap is obtained for hyperbolic
polyhedral surfaces, homeomorphic to the double torus, whose 1-skeletons are
cubic graphs.
