Using density functional theory calculations, we have studied the morphology of a Pt-37 nanoparticle supported on carbon with and without hydrogen (H) passivation that arises with postprocessing of nanoparticles before characterization. Upon heating in an anneal cycle, we find that without H (e.g., in a helium atmosphere or evacuation at high temperature), the morphology change of a truncated cuboctahedral Pt-37 is driven by the shearing of (100) to (111) facets to lower the surface energy, a remnant shear instability that drives surface reconstruction in semi-infinite Pt(100). With H passivation from a postprocessing anneal, we show that the sheared structure automatically reverts to the observed truncated cuboctahedral structure and the average first nearest-neighbor Pt-Pt bond length increases by 3%, agreeing well with experiment. We explain the stabilization of the truncated cuboctahedral structure due to H passivation via adsorption energetics of hydrogen on Pt(100) and (111) facets, specifically, the preference for H adsorption at bridge sites on (100) facets, which should be considered in a realistic model for H adsorption on Pt nanoparticles. We find that dramatic morphological change of a nanoparticle can occur even with small changes to first-shell Pt-Pt coordination number. The implications of our findings when comparing to experimental data are discussed.