Polytropes have gained renewed interest because they account for several seemingly disconnected observational properties of galaxies. Here we study whether polytropes are also able to explain the stellar mass distribution within galaxies. We develop a code to fit surface density profiles using polytropes projected in the plane of the sky (propols). Sérsic profiles are known to be good proxies for the global shapes of galaxies and we find that, ignoring central cores, propols, and Sérsic profiles are indistinguishable within observational errors (within 5% over five orders of magnitude in surface density). The range of physically meaningful polytropes yields Sérsic indexes between 0.4 and 6. The code has been systematically applied to ~750 galaxies with carefully measured mass density profiles and including all morphological types and stellar masses ( $7\lt {\rm{log}}[{M}_{\star }/{{\rm{M}}}_{\odot }]\lt 12$ ). The propol fits are systematically better than Sérsic profiles when ${\rm{log}}({M}_{\star }/{{\rm{M}}}_{\odot })\lesssim 9$ and systematically worse when ${\rm{log}}({M}_{\star }/{{\rm{M}}}_{\odot })\gtrsim 10$ . Although with large scatter, the observed polytropic indexes increase with increasing mass and tend to cluster around m = 5. For the most massive galaxies, propols are very good at reproducing their central parts, but they do not handle well cores and outskirts overall. Polytropes are self-gravitating systems in thermal meta-equilibrium as defined by the Tsallis entropy. Thus, the above results are compatible with the principle of maximum Tsallis entropy dictating the internal structure in dwarf galaxies and in the central region of massive galaxies.