The super-massive black holes of 10(6)M(⊙) to 10(9)M(⊙) that reside in the nuclei of active galaxies (AGN) are surrounded by a region emitting broad lines, probably associated with an accretion disk. The diameters of the broad-line regions range from a few light-days to more than a hundred light-days, and cannot be resolved spatially. The relative significance of inflow, outflow, rotational or turbulent motions in the broad-line regions as well as their structure (spherical, thin or thick accretion disk) are unknown despite intensive studies over more than thirty years. Here we report a fundamental relation between the observed emission linewidth full-width at half-maximum (FWHM) and the emission line shape FWHM/σ(line) in AGN spectra. From this relation we infer that the predominant motion in the broad-line regions is Keplerian rotation in combination with turbulence. The geometry of the inner region varies systematically with the rotation velocity: it is flattest for the fast-rotating broad-line objects, whereas slow-rotating narrow-line AGN have a more spherical structure. Superimposed is the trend that the line-emitting region becomes geometrically thicker towards the centre within individual galaxies. Knowing the rotational velocities, we can derive the central black-hole masses more accurately; they are two to ten times smaller than has been estimated previously.