The velocity coefficient (α) and its variation across the free surface in geometric channels have not been explicitly examined. This study employs the Single Point Method (SPM), a cost-effective and straightforward approach, to investigate α in channels with different bed and wall roughness conditions. Defined as the ratio of depth- averaged velocity (Ud) to surface velocity (us), α was determined at each free-surface point using a power-law vertical velocity distribution. Sectional α values were then computed as weighted averages for smooth boundaries and eight roughness combinations. The method was also applied to generate iso-velocity contours in trapezoidal, rectangular, and triangular sections. Validation against experimental data confirmed the high accuracy of SPM, with relative errors of less than 6%. Results show that as the channel narrows, the maximum velocity shifts away from the free surface and α increases (α=1.05 in a narrow rectangular channel with smooth boundaries). In wider channels, the maximum velocity approaches the free surface, reducing α (α=0.706 in a wide trapezoidal channel with smooth boundaries). The presence of rough solid boundaries (walls or bed) results in a systematic displacement of isovelocity contours away from the rougher channel surfaces, and modifies sectional α (0.658–1.12). These findings demonstrate that the commonly assumed default value of α = 0.85 is not universally applicable across all channel geometries. Comparisons with previous studies further confirm that the obtained α values fall within acceptable ranges, with relative errors under similar conditions ranging from 0.65% to 15.33%.

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