  The Turaev-Viro state sum invariant is known to give the transition amplitude
for the three dimensional BF theory with cosmological term, and its deformation
parameter hbar is related with the cosmological constant via hbar=sqrt{Lambda}.
This suggests a way to find the expectation value of the spacetime volume by
differentiating the Turaev-Viro amplitude with respect to the cosmological
constant. Using this idea, we find an explicit expression for the spacetime
volume in BF theory. According to our results, each labelled triangulation
carries a volume that depends on the labelling spins. This volume is explicitly
discrete. We also show how the Turaev-Viro model can be used to obtain the
spacetime volume for (2+1) dimensional quantum gravity.
