A simple direct proof is given of Minkowski's result that the mean length of the orthogonal projection of a convex set in E 3 onto an isotropic random line is (2 π ) –1 times the integral of mean curvature over its surface. This proof is generalised to a correspondingly direct derivation of an analogous formula for the mean projection of a convex set in E n onto an isotropic random s -dimensional subspace in E n . (The standard derivation of this, and a companion formula, to be found in Bonnesen and Fenchel's classic book on convex sets, is most indirect.) Finally, an alternative short inductive derivation (due to Matheron) of both formulae, by way of Steiner's formula, is presented.