The steric repulsion exerted on spherical particles by a semidilute polymer brush is evaluated using numerical self-consistent field theory (SCFT) in cylindrical coordinates. This accurate treatment provides the opportunity to test the conventional analytical approach, where the strong-stretching theory (SST) of Milner, Witten, and Cates for uniform compression can be adapted to particles of any shape using the Derjaguin approximation. While the Derjaguin approximation works well, at least, for the interaction energy, the SST proves to be seriously inaccurate for realistic grafting densities. Nevertheless, an efficient and accurate treatment for arbitrarily shaped particles remains possible if the uniform compression in the Derjaguin approximation is supplied by SCFT.