Surface brightnesses have been obtained for 52 southern Cepheids and 63 northern Cepheids using the (V-R) color index. A period-radius relation (in the period range of 3-45 days) of log R = 1.108 + or - 0.743 log P is obtained, with uncertainties of + or - 0.023 in the zero point and slope. The influence of unresolved binaries and possible overtone pulsators on this relation is considered. The present period-radius relation is shown to be consistent with current determinations of distances and effective temperature scales for classical Cepheids. No evidence is found to support the contention that all shorter period Cepheids are overtone pulsators.