Structures and electronic properties of rhombohedral [1 1 1] and [1 1 0] bismuth nanowires are calculated with the use of density functional theory. The formation of an energy band gap from quantum confinement is studied and to improve estimates for the band gap the GW approximation is applied. The [1 1 1] oriented nanowires require surface bonds to be chemically saturated to avoid formation of metallic surface states, whereas the surfaces of the [1 1 0] nanowires do not support metallic surface states. It is found that the onset of quantum confinement in the surface passivated [1 1 1] nanowires occurs at larger critical dimensions than for the [1 1 0] nanowires. For the [1 1 1] oriented nanowires it is predicted that a band gap of ~0.5 eV can be formed at a diameter of approximately 6 nm, whereas for the [1 1 0] oriented nanowires a diameter of approximately 3 nm is required to achieve a similar band gap energy. The GW correction is also applied to estimates of the electron affinity, ionisation potentials and work functions for both orientations of the nanowires for various diameters below 5 nm. The magnitude of the energy band gaps that arise in bismuth at critical dimensions of a few nanometers are of the same order as for conventional bulk semiconductors.