We investigate the properties of circumplanetary discs formed in three-dimensional, self-gravitating radiation hydrodynamical models of gas accretion by protoplanets. We determine disc sizes, scaleheights, and density and temperature profiles for different protoplanet masses, in solar nebulae of differing grain opacities. We find that the analytical prediction of circumplanetary disc radii in an evacuated gap (RHill/3) from Quillen Trilling yields a good estimate for discs formed by high-mass protoplanets. The radial density profiles of the circumplanetary discs may be described by power laws between r-2 and r-3/2. We find no evidence for the ring-like density enhancements that have been found in some previous models of circumplanetary discs. Temperature profiles follow a a?? r-7/10 power law regardless of protoplanet mass or nebula grain opacity. The discs invariably have large scaleheights (H/r > 0.2), making them thick in comparison with their encompassing circumstellar discs, and they show no flaring.