Confidence region procedures for multidimensional quantities sometimes require prohibitive amounts of computation and the regions are difficult to represent in a useful way. Some approximate procedures are constructed by using regions obtained as the intersection of several regions, each much easier to construct. The procedures are applicable to the solution of simultaneous equations, whose coefficients are subject to random error. Approximations by convex polyhedra and by parallelepipeds are proposed. The procedures are illustrated for setting a confidence region for the location of the vertex of a quadratic regression surface.