Choose $n$ independent random points on the boundary of a convex body $K \subset \R^d$. The intersection of the supporting halfspaces at these random points is a random convex polyhedron. The expectations of its volume, its surface area and its mean width are investigated. In the case that the boundary of $K$ is sufficiently smooth, asymptotic expansions as $n \to \infty$ are derived even in the case when the curvature is allowed to be zero. We compare our results to the analogous results for best approximating polytopes.