We report the study of the far-infrared (IR) sizes of submillimeter galaxies (SMGs) in relation to their dust-obscured star formation rate (SFR) and active galactic nuclei (AGN) presence, determined using mid-IR photometry. We determined the millimeter-wave ({lambda}_obs_=1100um) sizes of 69 Atacama Large Millimeter/submillimeter Array (ALMA)-identified SMGs, selected with >=10{sigma} confidence on ALMA images (F_1100um_=1.7-7.4mJy). We found that all of the SMGs are located above an avoidance region in the size-flux plane, as expected by the Eddington limit for star formation. In order to understand what drives the different millimeter-wave sizes in SMGs, we investigated the relation between millimeter-wave size and AGN fraction for 25 of our SMGs at z=1-3. We found that the SMGs for which the mid-IR emission is dominated by star formation or AGN have extended millimeter-sizes, with respective median R_c,e_=1.6_-0.21_^+0.34^ and 1.5_-0.24_^+0.93^kpc. Instead, the SMGs for which the mid-IR emission corresponds to star-forming/AGN composites have more compact millimeter-wave sizes, with median R_c,e_=1.0_-0.20_^+0.20^kpc. The relation between millimeter-wave size and AGN fraction suggests that this size may be related to the evolutionary stage of the SMG. The very compact sizes for composite star-forming/AGN systems could be explained by supermassive black holes growing rapidly during the SMG coalescing, star-formation phase.