The ratio of maximum radial field to maximum axial field has been calculated numerically for axisymmetric solenoids of rectangular cross section and constant current density over a wide range of coil configurations. It is also shown that for a certain range of coil shapes the maximum modulus of the magnetic field occurs on the end faces of the windings. These considerations are pertinent in the design of superconducting magnets for two reasons: (i) Many superconductors have lower critical current in a field normal to the wide surface than in a field parallel to the surface, and (ii) large radial fields can cause catastrophic flux jumps.