For a Borel set A and a homogeneous Poisson point process λ in Rd of intensity λ > 0, define the Poisson-Voronoi approximation Aλ of A as a union of all Voronoi cells with nuclei from λ lying in A. If A has a finite volume and perimeter, we find an exact asymptotic of E Vol(AδAη) as λ→∞, where Vol is the Lebesgue measure. Estimates for all moments of Vol(Aη) and Vol(AδAη) together with their asymptotics for large λ are obtained as well. © Applied Probability Trust 2012.