For a Borel set $A$ and a stationary Poisson point process $η_t$ in $\mathbb R^d$ of intensity $t>0$, the Poisson-Delaunay approximation $ A_{η_t}$ of $A$ is the union of all Delaunay cells generated by $η_t$ with center in $A$. It is shown that $λ_d(A_{η_t})$ is an unbiased estimator for $λ_d(A)$, variance bounds and a quantitative central limit theorem are given. The asymptotic behaviour of the symmetric difference $λ_d(AΔA_{η_t})$ is derived as $t \to\infty$.