While you can measure it via scattering, it's not the square root of the scattering cross section. The radius is defined via the slope of the proton electric fo

While you can measure it via scattering, it's not the square root of the scattering cross section. The radius is defined via the slope of the proton electric form factor at Q^2=0. This translates to the root mean square radius, that is the sqrt of the integral of r^2 weighted with the density, i.e. sqrt ( Integral r^2 rho(r)).