  A puzzling discrepancy exists between the values of the proton charge radius
obtained using different experimental techniques: elastic electron-proton
scattering and spectroscopy of electronic and muonic hydrogen. The proton
radius is defined through the slope of the electric form factor, $G_E(Q^2)$, at
zero four-momentum transfer, which is inaccessible in scattering experiments.
We propose a novel method for extracting the proton radius from scattering data
over a broad $Q^2$ range rather than attempting to directly determine the slope
of $G_E$ at $Q^2 = 0$. This method relates the radius of the proton to its
transverse charge density, which is the two-dimensional Fourier transform of
the Dirac form factor, $F_1(Q^2)$. We apply our method to reanalyze the
extensive data obtained by the A1 Collaboration [J. C. Bernauer et al., Phys.
Rev. Lett. 105, 242001 (2010)] and extract a radius value, $r_E =
0.889(5)_{\text{stat}}(5)_{\text{syst}}(4)_{\text{model}}~\text{fm}$, that is
consistent with the original result. We also provide new parametrizations for
the Dirac and Pauli form factors and the transverse charge and magnetization
densities of the proton. Our reanalysis shows that the proton radius
discrepancy cannot be explained by issues with fitting and extrapolating the A1
data to $Q^2 = 0$.
