  Exploiting a very large library of physically plausible equations of state
(EOSs) containing more than $10^{7}$ members and yielding more than $10^{9}$
stellar models, we conduct a survey of the impact that a neutron-star radius
measurement via electromagnetic observations can have on the EOS of nuclear
matter. Such measurements are soon to be expected from the ongoing
\textit{Neutron Star Interior Composition Explorer} mission and will complement
the constraints on the EOS from gravitational-wave detections. Thanks to the
large statistical range of our EOS library, we can obtain a first quantitative
estimate of the commonly made assumption that the high-density part of the EOS
is best constrained when measuring the radius of the most massive, albeit rare,
neutron stars with masses $M\gtrsim2.1\,M_\odot$. At the same time, we find
that radius measurements of neutron stars with masses
$M\simeq1.7-1.85\,M_\odot$ can provide the strongest constraints on the
low-density part of the EOS. Finally, we quantify how radius measurements by
future missions can further improve our understanding of the EOS of matter at
nuclear densities.
