The authors investigate random polytopes that arise as the convex hull of random, independent points chosen from the boundary of a convex body with twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. In the past, much effort has been devoted to the study of random polytopes whose vertices are chosen from inside the convex body \(K\). The authors prove analogous results for the inscribed model. In particular, they obtain a lower bound on the variance of the volume of the random polytope. With this, they prove that the upper bound obtained by M. Reitzner [Ann. Probab. 31, No. 4, 2136–2166 (2003; Zbl 1058.60010)] is sharp. They also establish that the volume has an exponential tail. This result implies upper bounds on the higher moments of the volume of the random polytope. Finally, a central limit theorem is proved for a Poisson model whose expected volume is asymptotically equivalent to the inscribed models.