We present detailed Fourier modeling of the radio remnant of SN 1987A using high-resolution 9 and 18 GHz data taken with the Australia Telescope Compact Array over the period 1992-2008. We develop a parameterized three-dimensional torus model for the expanding radio shell, in which the emission is confined to an inclined equatorial belt; our model also incorporates both a correction for light-travel time effects and an overall east-west gradient in the radio emissivity. By deriving an analytic expression for the two-dimensional Fourier transform of the projected three-dimensional brightness distribution, we can fit our spatial model directly to the interferometric visibility data. This provides robust estimates of the radio morphology at each epoch. The best-fit results suggest a constant remnant expansion at 4000 ± 400 km s−1 over the 16 yr period covered by the observations. The model fits also indicate substantial midlatitude emission, extending to ±40° on either side of the equatorial plane. This likely corresponds to the extraplanar structure seen in Hα and Lyα emission from the supernova reverse shock, and broadly supports hydrodynamic models in which the complex circumstellar environment was produced by a progression of interacting winds from the progenitor. Our model quantifies the clear asymmetry seen in the radio images: we find that the eastern half of the radio remnant is consistently ~40% brighter than the western half at all epochs, which may result from an asymmetry in the ejecta distribution between these two hemispheres.