Manning equation: what is the flow?

Pick the channel shape and lining, then type its size, the flow depth and the slope. The Manning equation calculator gives the discharge and mean velocity of uniform flow, with the flow area, wetted perimeter and hydraulic radius.

Your numbers

UnitsUSMetric

Channel shape

TrapezoidRectangleTriangle (V)Circular pipe

Bottom width (b)

Side slope (z:1)

Flow depth (y)

Slope (S, ft/ft or m/m)

Lining

Discharge: Q = 165.44 ft³/s (4.6846 m³/s)

Manning’s equation gives Q = 165.44 ft³/s (4.6846 m³/s) at a mean velocity of 3.4466 ft/s.

Discharge (ft³/s): 165.44
Discharge (m³/s): 4.6846
Discharge (L/s): 4,684.6
Discharge (gal/min): 74,252
Velocity (ft/s): 3.4466
Velocity (m/s): 1.0505
Flow area (A): 48 ft²
Wetted perimeter (P): 23.416 ft
Hydraulic radius (R): 2.0498 ft
Top width (T): 22 ft

Discharge: Q = 165.44 ft³/s (4.6846 m³/s). Manning’s equation gives Q = 165.44 ft³/s (4.6846 m³/s) at a mean velocity of 3.4466 ft/s.

How to calculate

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.

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.

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.

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.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

Channel shape Rectangle, Bottom width (b) 6.562 ft, Flow depth (y) 3.281 ft, Slope (S, ft/ft or m/m) 0.001, Lining Concrete channel, trowel finish, 0.013 gives Velocity (m/s) 1.532392, Discharge (m³/s) 3.064785, Hydraulic radius (R) 1.6404 ft.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)
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 gives Discharge (ft³/s) 165.435057, Velocity (ft/s) 3.446564, Flow area (A) 48 ft².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)
Channel shape Circular pipe, Pipe diameter (D) 3.281 ft, Flow depth (y) 3.281 ft, Slope (S, ft/ft or m/m) 0.005, Lining Concrete pipe with bends and some debris, 0.013 gives Velocity (m/s) 2.158581, Discharge (m³/s) 1.695345, Hydraulic radius (R) 0.82021 ft, Top width (T) 0 ft.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)
Channel shape Circular pipe, Pipe diameter (D) 3.281 ft, Flow depth (y) 1.64 ft, Slope (S, ft/ft or m/m) 0.005, Lining Concrete pipe with bends and some debris, 0.013 gives Velocity (m/s) 2.158581, Discharge (m³/s) 0.847673, Top width (T) 3.2808 ft.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)
Channel shape Triangle (V), Side slope (z:1) 3, Flow depth (y) 1.64 ft, Slope (S, ft/ft or m/m) 0.01, Lining Earth with short grass, few weeds, 0.027 gives Discharge (m³/s) 1.064319, Flow area (A) 8.0729 ft².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)

How it works

Manning's equation in metres: V = (1 ÷ n) × R^(2/3) × S^(1/2), and Q = V × A, with R = A ÷ P. The page converts every length to metres first, so in feet it is the same as V = (1.4859 ÷ n) × R^(2/3) × S^(1/2), where 1.4859 = (1 ÷ 0.3048)^(1/3); manuals round it to 1.486.

Cross-sections (y is the flow depth):

Rectangle, bottom width b: A = b × y, P = b + 2y, top width T = b.
Trapezoid, bottom width b, side slopes z horizontal to 1 vertical: A = (b + z × y) × y, P = b + 2y√(1 + z²), T = b + 2zy.
Triangle (V), side slopes z: A = z × y², P = 2y√(1 + z²), T = 2zy.
Circular pipe, diameter D, partly full: the water surface cuts off a segment with central angle θ = 2 × arccos(1 − 2y ÷ D). Then A = D² × (θ − sin θ) ÷ 8, P = D × θ ÷ 2 and T = D × sin(θ ÷ 2) (0 when full). Full, A = πD² ÷ 4 and P = πD.

Manning's n (USACE HEC-RAS normal values, from Chow 1959): concrete pipe, straight and clean 0.011; concrete pipe with bends and some debris 0.013; cast iron pipe, coated 0.013; steel pipe, lockbar and welded 0.012; corrugated metal storm drain 0.024; Lucite (acrylic) pipe 0.009; concrete channel, trowel finish 0.013, float finish 0.015, unfinished 0.017; gunite, good section 0.019; asphalt, smooth 0.013; brick in cement mortar 0.015; earth, clean, recently completed 0.018; earth, clean, after weathering 0.022; earth, gravel, uniform and clean 0.025; earth with short grass, few weeds 0.027; natural stream, clean and straight 0.030; clean and winding 0.040; mountain stream, gravel and cobbles 0.040; natural stream, sluggish and weedy 0.070. Or type your own n. For a natural stream, the USGS photographs of 22 measured channels, with n from 0.024 to 0.079, help to pick a value.

Rules

Width, diameter and depth from 0.0001 m to 10,000 m; z from 0 to 100; slope from 10⁻⁷ to 1; n from 0.005 to 0.5.
In a pipe the depth cannot be more than the diameter; a depth within 10⁻¹² of the diameter (the same size typed in two units) counts as full. A triangle needs z above 0.
Flow that is 0 or not finite has no answer.

Output format

The headline reads Q = a ft³/s (b m³/s), each to 5 significant figures with thousands commas (a × 10ⁿ below 10⁻⁶ or from 10¹⁵ up). The discharge (ft³/s, m³/s, L/s, US gal/min), velocity (ft/s, m/s), area, wetted perimeter, hydraulic radius and top width show to 5 significant figures; the geometry in feet (metres in Metric). Every result is a 64-bit float (roots, powers and trigonometry).

Worked examples by hand

Concrete rectangle. b = 2 m, y = 1 m, S = 0.001, n = 0.013 (trowel finish): A = 2 m², P = 2 + 2 = 4 m, R = 0.5 m. V = 0.5^(2/3) × √0.001 ÷ 0.013 = 0.629961 × 0.0316228 ÷ 0.013 = 1.53239 m/s; Q = 3.06478 m³/s.

Earth trapezoid. b = 10 ft, z = 2, y = 3 ft, S = 0.001, n = 0.022: A = (10 + 2 × 3) × 3 = 48 ft², P = 10 + 6√5 = 23.4164 ft, R = 2.04984 ft. In metres R = 0.624793 m and V = 0.624793^(2/3) × √0.001 ÷ 0.022 = 1.05051 m/s = 3.44656 ft/s; Q = 3.44656 × 48 = 165.435 ft³/s. (With the rounded 1.486 it would be 3.44675 ft/s.)

Full concrete pipe. D = 1 m, y = 1 m, S = 0.005, n = 0.013: θ = 2π, A = π ÷ 4 = 0.785398 m², P = π, R = 0.25 m. V = 0.25^(2/3) × √0.005 ÷ 0.013 = 2.15858 m/s; Q = 1.69535 m³/s.

Half-full pipe. y = 0.5 m: θ = π, A = π ÷ 8, P = π ÷ 2, R = 0.25 m, so V is the same 2.15858 m/s and Q = 0.847673 m³/s, with a top width of 1 m.

Grassy V ditch. z = 3, y = 0.5 m, S = 0.01, n = 0.027: A = 3 × 0.25 = 0.75 m², P = 2 × 0.5 × √10 = 3.16228 m, R = 0.237171 m. V = 0.237171^(2/3) × 0.1 ÷ 0.027 = 1.41909 m/s; Q = 1.06432 m³/s.