On the assumption of a uniform charge distribution, the Coulomb energy constant obtained from a recent adjustment of the semiempirical formula to the mass data corresponds to the Coulomb radius constant 1.237 (in 10-13 cm). This result is in good agreement with the radius constant obtained from recent studies of μ-mesonic x-rays, and can be readily reconciled with the radii obtained from electron scattering, isotope shift measurements, and mirror nuclide mass differences. In view of the importance of the question of the Coulomb radius, an attempt is made to refine the determination of the Coulomb radius constant based upon nuclear masses by using an objective criterion for best fit. A least-squares analysis involving a new adjustment procedure yields the radius constant 1.216. An investigation of the precision of this determination leads to the assignment of a probable error of 1 percent. This new radius constant agrees with the average radius constant obtained from μ-mesonic x-rays within the small probable error assigned to each.