The application of the Manning equation to partly filled circular channels is considered. The four types of problem that require iterative calculations are elucidated, and explicit solutions proposed in the literature are reviewed. New and improved equations are presented for one type of problem. Explicit analytical equations are developed for the two types of problem for which no explicit solutions could be found in the literature. For each type of problem, two cases are considered: (i) constant Manning roughness coefficient, and (ii) variable Manning roughness coefficient that depends on the depth of flow. Separate equations are presented for each case. The accuracy of each equation is demonstrated by calculating and reporting its maximum deviation from the exact solution within its range of applicability. In addition to obviating the need for iterative calculations, these equations facilitate the calculation of both solutions when the problem at hand has two distinct solutions (two possible flow depths). The proposed equations are accurate enough to be used in computer calculations and sufficiently simple to be used with a hand-held calculator.