Part of this is also that a proton isn't a elementary particle. And in fact it's not only the three quarks typically ascribed; that's just only a little over 1

Part of this is also that a proton isn't a elementary particle. And in fact it's not only the three quarks typically ascribed; that's just only a little over 1% of it's mass. The rest is a maelstrom of virtual particles. Defining the radius anything more than extremely statistically runs up against the uncertainty principle of all these constituent particles.