We study 2D quantum gravity on spherical topologies employing the Regge cal
culus approach with the dl/l measure. Instead of the normally used fixed no
nregular triangulations we study random triangulations which are generated
by the standard Voronoi-Delaunay procedure. For each system size we average
the results over four different realizations of the random lattices. We co
mpare both types of triangulations quantitatively and investigate how the d
ifference in the expectation Value of the squared curvature, R-2, for fixed
and random triangulations depends on the lattice size and the surface area
A. We try to measure the string susceptibility exponents through finite-si
ze scaling analyses of the expectation value of an added R-2 interaction te
rm, using two conceptually quite different procedures. The approach, where
an ultraviolet cutoff is held fixed in the scaling limit, is found to be pl
agued with inconsistencies, as has already previously been pointed out by u
s. In a conceptually different approach, where the area A is held fixed, th
ese problems are not present. We find the string susceptibility exponent ga
mma'(str), in rough agreement with theoretical predictions for the sphere,
whereas the estimate for gamma(str), appears to be too negative. However, o
ur results are hampered by the presence of severe finite-size corrections t
o scaling, which lead to systematic uncertainties well above our statistica
l errors. We feel that the present methods of estimating the string suscept
ibilities by finite-size scaling studies are not accurate enough to serve a
s testing grounds to decide the success of failure of quantum Regge calculu
s.