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 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 \gtrsim 2.1 M _ { \odot } . At the same time , we find that radius measurements of neutron stars with masses M \simeq 1.7 - 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 .