Compute the hydraulic radius (R = A / P) for circular, rectangular, trapezoidal, and triangular sections. Visualize the cross‑section geometry with interactive canvas.
The hydraulic radius (R) is a fundamental geometric parameter in open‑channel flow and pipe flow hydraulics. Defined as the ratio of the cross‑sectional area of flow (A) to the wetted perimeter (P), it quantifies the efficiency of a channel or pipe in conveying fluid. A larger hydraulic radius indicates a more efficient section, with less frictional resistance per unit of flow area.
R = A⁄P
where A = cross‑sectional area of flow [L²], P = wetted perimeter [L]
The hydraulic radius appears in the Manning equation and the Chezy formula, two of the most widely used empirical relations for estimating flow velocity and discharge in open channels and closed conduits. It directly influences flow capacity, sediment transport, and energy losses. Engineers use it to design stormwater drains, irrigation canals, sewer systems, and culverts.
D = diameter
θ = 2 arccos(1 − 2h/D), h = water depth
b = bottom width, h = water depth
b = bottom width, h = depth, z = side slope (H:V)
h = water depth, z = side slope (H:V)
The concept of hydraulic radius originates from the need to simplify the complex physics of turbulent flow in open channels. By combining the cross‑sectional area and the wetted perimeter into a single parameter, engineers can compare the hydraulic efficiency of different shapes. The hydraulic radius is not a geometric radius in the conventional sense; rather, it is a characteristic length that represents the ratio of flow area to the boundary surface in contact with the fluid. For a circular pipe flowing full, R = D/4, which is half the actual radius — a result of the mathematical definition.
In the Manning equation, V = (1/n) R2/3 S1/2, the hydraulic radius appears with a 2/3 exponent, reflecting its strong influence on flow velocity. This equation is empirically derived and has been validated across thousands of field and laboratory measurements, making it a cornerstone of hydraulic engineering.
The following table shows pre‑computed values for standard sections, verified against textbook solutions (Chow, 1959; French, 1985).
| Shape | Dimensions | Area A | Wetted Perimeter P | Hydraulic Radius R |
|---|---|---|---|---|
| Circular (full) | D = 1.0 m | 0.7854 m² | 3.1416 m | 0.2500 m |
| Circular (partial) | D = 1.0 m, h = 0.6 m | 0.4917 m² | 1.7722 m | 0.2775 m |
| Rectangular | b = 2.0 m, h = 1.2 m | 2.4000 m² | 4.4000 m | 0.5455 m |
| Trapezoidal | b = 1.5 m, h = 1.0 m, z = 1.5 | 3.0000 m² | 5.1056 m | 0.5876 m |
| Triangular | h = 1.5 m, z = 1.0 | 2.2500 m² | 4.2426 m | 0.5303 m |
A civil engineer is designing a storm drain for a new residential development. The peak flow is estimated at 2.5 m³/s, and the available slope is 0.5%. Using the Manning equation with n = 0.013 (concrete pipe), the engineer needs to select the minimum pipe diameter that can convey the flow. By calculating the hydraulic radius for various diameters, the engineer determines that a 1.2‑m diameter circular pipe (full flow) provides R = 0.30 m and a velocity of 3.8 m/s, which is above the minimum self‑cleaning velocity of 0.6 m/s. The tool verifies this design by computing the area, perimeter, and hydraulic radius instantly, allowing rapid iteration.
Extended Insight: When applying this in practice, note that the Manning roughness coefficient n is not truly constant—it varies with depth and velocity. However, the hydraulic radius R remains the primary geometric control. Sensitivity analyses show that a 10% increase in R yields roughly a 6.5% increase in velocity (since V ∝ R2/3), underscoring the importance of precise geometric computation.
For a given area, the section with the minimum wetted perimeter (and thus the maximum hydraulic radius) is the most efficient. For example, among all trapezoidal sections, the half‑hexagon (with side slope z = 1/√3 ≈ 0.577 and bottom width b = 2h/√3) yields the highest R for a given depth. This principle is widely used in designing economical irrigation canals where lining costs are proportional to the wetted perimeter. While this tool focuses on direct computation, the interactive feedback helps users quickly identify near‑optimal dimensions for their specific constraints.
In natural streams and complex hydraulic structures, the cross‑section may consist of multiple subsections (e.g., main channel and floodplains) with different roughness coefficients. The overall hydraulic radius is computed by dividing the total area by the total wetted perimeter, but the flow capacity is determined by summing the contributions of each subsection. This calculator focuses on single‑section geometries, which are the building blocks for more advanced analyses.