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