  We study 2D quantum gravity on spherical topologies using the Regge calculus
approach with the $dl/l$ measure. Instead of a fixed non-regular triangulation
which has been used before, we study for each system size four different random
triangulations, which are obtained according to the standard Voronoi-Delaunay
procedure. We compare both approaches quantitatively and show that the
difference in the expectation value of $R^2$ between the fixed and the random
triangulation depends on the lattice size and the surface area $A$. We also try
again to measure the string susceptibility exponents through a finite-size
scaling Ansatz in the expectation value of an added $R^2$ interaction term in
an approach where $A$ is held fixed. The string susceptibility exponent
$\gamma_{str}'$ is shown to agree with theoretical predictions for the sphere,
whereas the estimate for $\gamma_{str}$ appears to be too negative.
