# Manning equation: what is the flow?

Manning equation calculator: velocity and discharge of uniform flow in a rectangular, trapezoidal, triangular or circular channel from its size, flow depth, slope and Manning roughness n.

- Page: https://www.acalculator.org/physics/manning-equation-calculator
- JSON spec: https://www.acalculator.org/physics/manning-equation-calculator.json
- Version: 9167e7631b67

## Default answer

Example with the default inputs (Channel shape Trapezoid, Bottom width (b) 10 ft, Side slope (z:1) 2, Flow depth (y) 3 ft, Slope (S, ft/ft or m/m) 0.001, Lining Earth, clean, after weathering, 0.022): Manning’s equation gives Q = 165.44 ft³/s (4.6846 m³/s) at a mean velocity of 3.4466 ft/s.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| sh | Channel shape | The shape of the channel or pipe cross-section. |
| b | Bottom width (b) | The width of the channel bed. |
| z | Side slope (z:1) | Horizontal run of each side wall per unit of height, for example 2 for 2:1. |
| D | Pipe diameter (D) | The inside diameter of the pipe. |
| y | Flow depth (y) | The depth of water at the deepest point. |
| S | Slope (S, ft/ft or m/m) | The fall of the channel bed per unit of length, for example 0.001 for 1 in 1,000 (0.1%). |
| mat | Lining | The channel or pipe material, which sets Manning’s n (USACE HEC-RAS, normal values). |
| n | Manning’s n | The roughness coefficient of the channel. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| result | Discharge | Q = V × A, in cubic feet and cubic metres per second. |
| cfs | Discharge (ft³/s) | The flow rate in cubic feet per second. |
| cms | Discharge (m³/s) | The flow rate in cubic metres per second. |
| lps | Discharge (L/s) | The flow rate in litres per second. |
| gpm | Discharge (gal/min) | The flow rate in US gallons per minute. |
| fps | Velocity (ft/s) | The mean velocity V = (k ÷ n) × R^(2/3) × S^(1/2), in feet per second. |
| mps | Velocity (m/s) | The mean velocity in metres per second. |
| area | Flow area (A) | The area of the water in the cross-section. |
| perimeter | Wetted perimeter (P) | The length of channel wall the water touches. |
| radius | Hydraulic radius (R) | R = A ÷ P. |
| top | Top width (T) | The width of the water surface. |

## Method

V = (1 ÷ n) × R^(2/3) × S^(1/2) in metres (1.486 ÷ n in feet), Q = V × A, R = A ÷ P; rectangle A = by, P = b + 2y; trapezoid A = (b + zy)y, P = b + 2y√(1 + z²); triangle A = zy², P = 2y√(1 + z²); pipe θ = 2 arccos(1 − 2y ÷ D), A = D²(θ − sin θ) ÷ 8, P = Dθ ÷ 2.

## Assumptions

- Steady, uniform flow: the depth and velocity stay the same along the channel, and the water surface is parallel to the bed.
- n is the normal value for the lining; real channels vary between the table’s minimum and maximum.

## Worked examples

1. sh = rect, b = 2, y = 1, S = 0.001, mat = trowel gives mps = 1.532392, cms = 3.064785, radius = 0.5. Source: USACE HEC-HMS Technical Reference Manual, Kinematic Wave Transform Model (Q = (K ÷ N) × R^(2/3) × A × √S, K = 1.486 or 1.0 in English or metric units; trapezoid wetted perimeter W + 2Y√(1 + Z²)), https://www.hec.usace.army.mil/confluence/hmsdocs/hmstrm/transform/kinematic-wave-transform-model (retrieved 2026-10-18); USACE HEC-RAS Hydraulic Reference Manual, Table 3-1 Manning’s n Values (Chow, 1959), https://www.hec.usace.army.mil/confluence/rasdocs/ras1dtechref/latest/basic-data-requirements/geometric-data/energy-loss-coefficients (retrieved 2026-10-18) (trowel-finished concrete, 0.013).
2. sh = trap, b = 3.048, z = 2, y = 0.9144, S = 0.001, mat = earth gives cfs = 165.435057, fps = 3.446564, area = 4.459346. Source: USACE HEC-HMS Technical Reference Manual, Kinematic Wave Transform Model (Q = (K ÷ N) × R^(2/3) × A × √S, K = 1.486 or 1.0 in English or metric units; trapezoid wetted perimeter W + 2Y√(1 + Z²)), https://www.hec.usace.army.mil/confluence/hmsdocs/hmstrm/transform/kinematic-wave-transform-model (retrieved 2026-10-18); USACE HEC-RAS Hydraulic Reference Manual, Table 3-1 Manning’s n Values (Chow, 1959), https://www.hec.usace.army.mil/confluence/rasdocs/ras1dtechref/latest/basic-data-requirements/geometric-data/energy-loss-coefficients (retrieved 2026-10-18) (earth, clean, after weathering, 0.022).
3. sh = circle, D = 1, y = 1, S = 0.005, mat = concretePipeBends gives mps = 2.158581, cms = 1.695345, radius = 0.25, top = 0. Source: USACE HEC-HMS Technical Reference Manual, Kinematic Wave Transform Model (Q = (K ÷ N) × R^(2/3) × A × √S, K = 1.486 or 1.0 in English or metric units; trapezoid wetted perimeter W + 2Y√(1 + Z²)), https://www.hec.usace.army.mil/confluence/hmsdocs/hmstrm/transform/kinematic-wave-transform-model (retrieved 2026-10-18); USACE HEC-RAS Hydraulic Reference Manual, Table 6-1 Manning’s n for Closed Conduits Flowing Partly Full (Chow, 1959), https://www.hec.usace.army.mil/confluence/rasdocs/ras1dtechref/latest/modeling-culverts/culvert-data-and-coefficients/manning-s-roughness-coefficient (retrieved 2026-10-18) (concrete culvert with bends, 0.013).
4. sh = circle, D = 1, y = 0.5, S = 0.005, mat = concretePipeBends gives mps = 2.158581, cms = 0.847673, top = 1. Source: USACE HEC-HMS Technical Reference Manual, Kinematic Wave Transform Model (Q = (K ÷ N) × R^(2/3) × A × √S, K = 1.486 or 1.0 in English or metric units; trapezoid wetted perimeter W + 2Y√(1 + Z²)), https://www.hec.usace.army.mil/confluence/hmsdocs/hmstrm/transform/kinematic-wave-transform-model (retrieved 2026-10-18).
5. sh = tri, z = 3, y = 0.5, S = 0.01, mat = grass gives cms = 1.064319, area = 0.75. Source: USACE HEC-HMS Technical Reference Manual, Kinematic Wave Transform Model (Q = (K ÷ N) × R^(2/3) × A × √S, K = 1.486 or 1.0 in English or metric units; trapezoid wetted perimeter W + 2Y√(1 + Z²)), https://www.hec.usace.army.mil/confluence/hmsdocs/hmstrm/transform/kinematic-wave-transform-model (retrieved 2026-10-18); USACE HEC-RAS Hydraulic Reference Manual, Table 3-1 Manning’s n Values (Chow, 1959), https://www.hec.usace.army.mil/confluence/rasdocs/ras1dtechref/latest/basic-data-requirements/geometric-data/energy-loss-coefficients (retrieved 2026-10-18) (short grass, 0.027).

## FAQ

### What is the Manning equation?

V = (k ÷ n) × R^(2/3) × S^(1/2), with V the mean velocity, n the roughness, R the hydraulic radius (flow area ÷ wetted perimeter) and S the slope. k is 1 in metres and 1.486 in feet. The discharge is Q = V × A.

### How do you calculate flow in a channel with the Manning equation?

Find the flow area A and the wetted perimeter P from the shape and depth, then R = A ÷ P. A concrete channel 2 m wide and 1 m deep has A = 2 m², P = 4 m and R = 0.5 m; with n = 0.013 and S = 0.001, V = 0.5^(2/3) × √0.001 ÷ 0.013 = 1.53 m/s and Q = 3.06 m³/s.

### What is a typical Manning n value?

About 0.011 to 0.013 for concrete pipe, 0.013 to 0.017 for concrete channels, 0.018 to 0.027 for earth channels and 0.030 to 0.070 for natural streams (USACE HEC-RAS, from Chow 1959). Rougher and weedier channels have higher n.

### Why is the US constant 1.486?

The equation was set up in metres. Rewriting it for feet multiplies by (1 ÷ 0.3048)^(1/3) = 1.4859, which manuals round to 1.486 or 1.49. This page works in metres, so its feet results use the exact 1.4859.

### Does a pipe carry the most water when it is full?

No. With these formulas a circular pipe carries the most, about 7.6% more than full, at about 94% of its depth, because the last bit of depth adds more wetted perimeter than area. Half full, it carries exactly half the full flow at the same velocity.

### What slope should I use?

The fall of the channel bed per unit length, as a ratio: a drop of 1 ft in 1,000 ft is 0.001 (0.1%). For uniform flow the water surface and the energy line have the same slope.

## Sources

- USACE, HEC-HMS Technical Reference Manual, Kinematic Wave Transform Model, Basic Concepts and Equations: Q = (K ÷ N) × R^(2/3) × A × √S with K = 1.486 in English units and 1.0 in metric units; Solution of Equations: trapezoid wetted perimeter W + 2Y√(1 + Z²), retrieved 2026-10-18. https://www.hec.usace.army.mil/confluence/hmsdocs/hmstrm/transform/kinematic-wave-transform-model
- USACE, HEC-RAS Hydraulic Reference Manual, Energy Loss Coefficients: Table 3-1, Manning’s n values for channels (from Chow, Open-Channel Hydraulics, 1959), retrieved 2026-10-18. https://www.hec.usace.army.mil/confluence/rasdocs/ras1dtechref/latest/basic-data-requirements/geometric-data/energy-loss-coefficients
- USACE, HEC-RAS Hydraulic Reference Manual, Manning’s Roughness Coefficient: Table 6-1, Manning’s n for closed conduits flowing partly full (Chow, 1959), retrieved 2026-10-18. https://www.hec.usace.army.mil/confluence/rasdocs/ras1dtechref/latest/modeling-culverts/culvert-data-and-coefficients/manning-s-roughness-coefficient
- USGS, Surface-Water Field Techniques, Verified Roughness Characteristics of Natural Channels (from Water-Supply Paper 1849, H. H. Barnes Jr.): photographs and measured n of 22 natural channels, 0.024 to 0.079, retrieved 2026-10-19. https://wwwrcamnl.wr.usgs.gov/sws/fieldmethods/Indirects/nvalues/index.htm
- Wolfram MathWorld, Circular Segment: segment area ½R²(θ − sin θ) and arc length Rθ, retrieved 2026-10-18. https://mathworld.wolfram.com/CircularSegment.html
- NIST Special Publication 811 (2008), Appendix B.8: 1 ft = 0.3048 m, 1 ft³ = 0.028316846592 m³, 1 US gallon = 3.785411784 L, retrieved 2026-10-18. https://www.nist.gov/pml/special-publication-811