  The mass radius is a fundamental property of the proton that so far has not
been determined from experiment. Here we show that the mass radius of the
proton can be rigorously defined through the formfactor of the trace of the
energy-momentum tensor (EMT) of QCD in the weak gravitational field
approximation, as appropriate for this problem. We then demonstrate that the
scale anomaly of QCD enables the extraction of the formfactor of the trace of
the EMT from the data on threshold photoproduction of $J/\psi$ and $\Upsilon$
quarkonia, and use the recent GlueX Collaboration data to extract the r.m.s.
mass radius of the proton ${\rm R_m = 0.55 \pm 0.03 \ fm}$. The extracted mass
radius is significantly smaller than the charge radius of the proton ${\rm R_C
= 0.8409 \pm 0.0004 \ fm}$. We attribute this difference to the interplay of
asymptotic freedom and spontaneous breaking of chiral symmetry in QCD, and
outline future measurements needed to determine the mass radius more precisely.
