We present a \textcolor blacksub-arcsecond resolution survey of the 340 GHz dust continuum emission from 50 nearby protoplanetary disks , based on new and archival observations with the Submillimeter Array . The observed visibility data were modeled with a simple prescription for the radial surface brightness profile . The results were used to extract intuitive , empirical estimates of the emission “ size ” for each disk , R _ { eff } , defined as the radius that encircles a fixed fraction of the total continuum luminosity , L _ { mm } . We find a significant correlation between the sizes and luminosities , such that R _ { eff } \propto L _ { mm } ^ { 0.5 } , providing a confirmation and quantitative characterization of a putative trend that was noted previously . This correlation suggests that these disks have roughly the same average surface brightness interior to their given effective radius , \sim 0.2 Jy arcsec ^ { -2 } ( or 8 K in brightness temperature ) . The same trend remains , but the 0.2 dex of dispersion perpendicular to this relation essentially disappears , when we account for the irradiation environment of each disk with a crude approximation of the dust temperatures based on the stellar host luminosities . We consider two ( not mutually exclusive ) explanations for the origin of this size–luminosity relationship . Simple models of the growth and migration of disk solids can account for the observed trend for a reasonable range of initial conditions , but only on timescales that are much shorter than the nominal ages present in the sample . An alternative scenario invokes optically thick emission concentrated on unresolved scales , with filling factors of a few tens of percent , that are perhaps manifestations of localized particle traps .