A new co-ordinate transformation describing concentric toroids enables the potential distribution of a toroidal concentric space-charge-limited diode to be calculated. The results are represented approximately by an analytic function that allows the calculation of perveance, current and other characteristics of a hollow-beam electron gun. It is shown that previous calculations are in error. The shapes of the focusing electrodes for a hollow-beam gun are determined by an electrolytic-tank analogue. It is thus possible to design a gun for thick hollow beams. It is shown that, for thick beams, the correct focusing electrodes for a hollow-beam gun are very different from those of a strip-beam gun. It is, however, possible, within a certain range of the relevant parameters, to use the Langmuir-Blodgett potentials of a cylindrical space-charge-limited diode in order to determine the ‘toroidal’ focusing electrodes in the electrolytic tank for the design of a hollow-beam gun.The potential distribution for a charge-free toroidal diode is measured and calculated. The toroidal diode theory includes the theory of cylindrical and spherical diodes as special cases. The results give us a method by which we can introduce a further correction, so that a hollow-beam gun for very thick beams can be designed. An experimental gun confirms these calculations.