Using data from the Sloan Digital Sky Survey (SDSS)-DR7, including structural measurements from 2D surface brightness fits with gim2d, we show how the fraction of quiescent galaxies depends on galaxy stellar mass M_*, effective radius R_e, fraction of r-band light in the bulge, B/T, and their status as a central or satellite galaxy at 0.01�<�z�<�0.2. For central galaxies we confirm that the quiescent fraction depends not only on stellar mass, but also on R_e. The dependence is particularly strong as a function of , with α�∼�1.5. This appears to be driven by a simple dependence on B/T over the mass range 9�<�log (M_*/M_⊙)�<�11.5, and is qualitatively similar even if galaxies with B/T�>�0.3 are excluded. For satellite galaxies, the quiescent fraction is always larger than that of central galaxies, for any combination of M_*, R_e and B/T. The quenching efficiency is not constant, but reaches a maximum of ∼0.7 for galaxies with 9�<�log (M_*/M_⊙)�<�9.5 and R_e�<�1�kpc. This is the same region of parameter space in which the satellite fraction itself reaches its maximum value, suggesting that the transformation from an active central galaxy to a quiescent satellite is associated with a reduction in R_e due to an increase in dominance of a bulge component.