Rotation periods obtained with the Kepler satellite have been combined with precise measurements of projected rotation velocity from the WIYN 3.5-m telescope to determine the distribution of projected radii for several hundred low-mass (0.1<=M/M_{sun}_<=0.8), fast-rotating members of the Pleiades cluster. A maximum likelihood modelling technique, that takes account of observational uncertainties, selection effects and censored data, and considers the effects of differential rotation and unresolved binarity, has been used to find that the average radius of these stars is 14+/-2 per cent larger at a given luminosity than predicted by current evolutionary models of Dotter et al. and Baraffe et al. The same models are a reasonable match to the interferometric radii of older, magnetically inactive field M dwarfs, suggesting that the over-radius may be associated with the young, magnetically active nature of the Pleiades objects. No evidence is found for any change in this over-radius above and below the boundary marking the transition to full convection. Published evolutionary models that incorporate either the effects of magnetic inhibition of convection or the blocking of flux by dark star-spots do not individually explain the radius inflation, but a combination of the two effects might. The distribution of projected radii is consistent with the adopted hypothesis of a random spatial orientation of spin axes; strong alignments of the spin vectors into cones with an opening semi-angle <30{deg} can be ruled out. Any plausible but weaker alignment would increase the inferred over-radius.