ABSTRACT We measure the projected half-light radii of young star clusters in 31 galaxies from the Legacy Extragalactic UV Survey (LEGUS). We implement a custom pipeline specifically designed to be robust against contamination, which allows us to measure radii for 6097 clusters. This is the largest sample of young star cluster radii currently available. We find that most (but not all) galaxies share a common cluster radius distribution, with the peak at around 3 pc. We find a clear mass–radius relation of the form Reff ∝ M0.24. This relation is present at all cluster ages younger than 1 Gyr, but with a shallower slope for clusters younger than 10 Myr. We present simple toy models to interpret these age trends, finding that high-mass clusters are more likely to be not tidally limited and expand. We also find that most clusters in LEGUS are gravitationally bound, especially at older ages or higher masses. Lastly, we present the cluster density and surface density distributions, finding a large scatter that appears to decrease with cluster age. The youngest clusters have a typical surface density of 100$\, \mathrm{ M}_\odot \, \mathrm{pc}^{-2}$.