  We compute the sum over flat surfaces of disc topology with arbitrary number
of conical singularities. To that end, we explore and generalize a specific
case of the matrix model of dually weighted graphs (DWG) proposed and solved by
one of the authors, M. Staudacher and Th. Wynter. Namely, we compute the sum
over quadrangulations of the disc with certain boundary conditions, with
parameters controlling the number of squares (area), the length of the boundary
and the coordination numbers of vertices. The vertices introduce conical
defects with angle deficit given by a multiple of $\pi$, corresponding to
positive, zero or negative curvature. Our results interpolate between the
well-known 2d quantum gravity solution for the disc with fluctuating 2d metric
and the regime of 'almost flat' surfaces with all the negative curvature
concentrated on the boundary. We also speculate on possible ways to study the
fluctuating 2d geometry with $AdS_2$ background instead of the flat one.
