Hydraulic Radius Calculator
Enter the geometry of your channel or pipe to get the hydraulic radius, hydraulic diameter, and - if you add a slope and roughness coefficient - the Manning velocity and discharge. Choose from five cross-section shapes: rectangular, trapezoidal, triangular, full circular pipe, or partially filled circular pipe. Switch between metric and imperial units at any time.
Formula
Worked example
A rectangular channel 3 m wide with 1.2 m water depth: A = 3 x 1.2 = 3.6 m², P = 3 + 2*1.2 = 5.4 m, R = 3.6 / 5.4 = 0.6667 m, D_h = 2.6667 m. With S = 0.001 and n = 0.013: V = (1/0.013) x 0.6667^(2/3) x 0.001^(1/2) = 2.03 m/s, Q = 2.03 x 3.6 = 7.31 m^3/s.
What is hydraulic radius?
The hydraulic radius (R) is the ratio of the cross-sectional flow area (A) to the wetted perimeter (P) of a channel or conduit: R = A / P. It is one of the most important parameters in open-channel hydraulics because it captures how efficiently a channel cross-section conveys flow. A larger hydraulic radius means more flow area per unit of friction surface, which translates to higher velocities and greater discharge for the same slope and roughness. For a full circular pipe of diameter D, the hydraulic radius simplifies to D/4, which is why the hydraulic diameter D_h = 4R = D for a full pipe and 4R for any other shape.
Shape-specific formulas
Each channel shape has its own area and wetted perimeter formula. For a rectangular channel of width b and depth y: A = b*y and P = b + 2y. For a trapezoidal channel with side slope z (horizontal per vertical): A = (b + z*y)*y and P = b + 2y*sqrt(1 + z^2). For a triangular channel (b = 0): A = z*y^2 and P = 2y*sqrt(1 + z^2). For a full circular pipe of radius r: A = pi*r^2 and P = 2*pi*r, giving R = r/2 = D/4. For a partially filled circular pipe with fill depth h, the central angle theta = 2*arccos((r - h)/r), then A = (r^2/2)*(theta - sin(theta)) and P = r*theta. Dividing in each case gives the hydraulic radius R.
Manning's equation and how hydraulic radius is used
Manning's equation is the workhorse of open-channel and gravity-pipe design. In SI units: V = (1/n)*R^(2/3)*S^(1/2), and discharge Q = V*A. In U.S. customary units the constant 1 becomes 1.486 (sometimes approximated as 1.49). Here n is Manning's roughness coefficient, a dimensionless measure of boundary friction (lower for smooth concrete, higher for rough natural channels), and S is the dimensionless channel slope (drop in invert elevation per unit length). Because R appears to the 2/3 power, it has a strong influence on velocity: doubling the hydraulic radius increases velocity by roughly 59%. Engineers adjust shape, size, and slope to meet both a minimum self-cleaning velocity (often 0.6-1.0 m/s for sewers) and a maximum non-erosive velocity (often 3-6 m/s depending on lining).
Hydraulic radius vs. hydraulic diameter
The hydraulic diameter D_h = 4R is often preferred over hydraulic radius in pressure-pipe and heat-transfer calculations because the Darcy-Weisbach equation and the Reynolds number formula are originally written for circular pipes with D as the pipe diameter. For a full circle, D_h = 4*(D/4) = D, so the two are consistent. For a rectangular duct of width W and height H, D_h = 4*W*H / (2*(W+H)) = 2*W*H / (W+H). Always check which parameter a formula expects: most open-channel formulas use R, while pressure-pipe friction and heat-transfer correlations use D_h. This calculator provides both so you can use the correct one for each application.
Typical Manning's n roughness coefficients
| Material / channel type | Manning's n range | Typical use |
|---|---|---|
| Smooth concrete pipe | 0.010-0.013 | Storm drains, sewers |
| Precast concrete pipe | 0.011-0.015 | Drainage culverts |
| Cast iron pipe | 0.011-0.015 | Water mains |
| Corrugated metal pipe | 0.019-0.027 | Culverts, storm drainage |
| Plastic (PVC, HDPE) | 0.009-0.013 | Sewers, drainage |
| Unlined rock tunnel | 0.025-0.040 | Headrace tunnels |
| Excavated earth (clean) | 0.018-0.025 | Drainage canals |
| Grassed channel | 0.025-0.040 | Agricultural drainage |
| Natural stream (clean) | 0.025-0.035 | Rivers, creeks |
| Natural stream (weedy) | 0.035-0.070 | Overgrown channels |
Common values used in open-channel and pipe flow calculations. Actual values vary with condition, age, and alignment.
Frequently asked questions
What is the hydraulic radius formula?
Hydraulic radius R = A / P, where A is the cross-sectional area of flow and P is the wetted perimeter - the length of the channel boundary that is in contact with the flowing water. For a full circular pipe of diameter D, this simplifies to R = D/4.
What is the difference between hydraulic radius and hydraulic diameter?
Hydraulic diameter D_h = 4R. For a full circular pipe, D_h equals the actual pipe diameter D, which is why the hydraulic diameter was defined this way - to make it consistent with the pipe diameter in standard friction factor and Reynolds number formulas. For non-circular cross-sections, always check which quantity a particular formula calls for.
What are typical Manning's n values for concrete pipes?
Smooth concrete pipe typically has a Manning's n of 0.010 to 0.013. Precast concrete pipe is slightly rougher at 0.011 to 0.015. These values can increase with age, joint misalignment, and sediment deposition, so a design should account for the expected service-life condition.
Why does a circular pipe not flow fastest when completely full?
A partially filled circular pipe (at roughly 93% full by depth) actually achieves the highest flow velocity, and maximum discharge occurs at about 94% full. This happens because at full flow the wetted perimeter increases faster than the area, reducing the hydraulic radius and therefore Manning's velocity. This is why sewer design standards often target 75-85% of the full-pipe discharge at peak design flow.
What is the most hydraulically efficient channel shape?
A semicircle is theoretically the most efficient open-channel shape because it maximises flow area for a given wetted perimeter, giving the highest hydraulic radius and thus the fastest flow per unit of excavation. In practice, trapezoidal channels are preferred because they are easier to construct. A trapezoidal channel with a side slope of 1:sqrt(3) (approximately 60 degree banks) approaches the semicircular ideal.
How do I find hydraulic radius for a partially filled pipe?
First calculate the central angle theta = 2*arccos((r - h)/r), where r is the pipe radius and h is the fill depth from the invert. Then the flow area A = (r^2 / 2)*(theta - sin(theta)) and the wetted perimeter P = r*theta. The hydraulic radius is R = A / P. This calculator performs all of those steps automatically from the diameter and fill depth.