AbstractLetK ⊂ Rdbe a convex body and choose pointsxl, x2, …, xnrandomly, independently, and uniformly fromK. ThenKn= conv {x1, …, xn} is a random polytope that approximatesK(asn→ ∞) with high probability. Answering a question of Rolf Schneider we determine, up to first order precision, the expectation of volK– volKnwhenKis a smooth convex body. Moreover, this result is extended to quermassintegrals (instead of volume).