  We extract the proton magnetic radius from the high-precision electron-proton
elastic scattering cross section data. Our theoretical framework combines
dispersion analysis and chiral effective field theory and implements the
dynamics governing the shape of the low-$Q^2$ form factors. It allows us to use
data up to $Q^2\sim$ 0.5 GeV$^2$ for constraining the radii and overcomes the
difficulties of empirical fits and $Q^2 \rightarrow 0$ extrapolation. We obtain
a magnetic radius $r_M^p$ = 0.850 $\pm$0.001 (fit 68%) $\pm$0.010 (theory full
range) fm, significantly different from earlier results obtained from the same
data, and close to the extracted electric radius $r_E^p$ = 0.842 $\pm$0.002
(fit) $\pm$0.010 (theory) fm.
