Manning's Equation Calculator

Manning’s equation is the workhorse formula of open-channel hydraulics. Every drainage ditch, every storm sewer entry, every irrigation canal, and most natural-stream models are sized using it. The form below assumes steady, uniform flow with a free water surface open to the atmosphere (not pressurized pipe flow).

The metric form:

V = (1/n) × R^(2/3) × S^(1/2)

Where V is mean flow velocity in m/s, n is Manning’s roughness coefficient (dimensionless), R is hydraulic radius in m, and S is channel slope in m/m. Multiply V by the cross-sectional area A to get flow rate Q = V·A.

The US-customary form:

V = (1.486/n) × R^(2/3) × S^(1/2)

V in ft/s, R in ft. The 1.486 conversion factor folds in the imperial unit shift. n stays the same (it’s dimensionless).

Where Manning’s n comes from:

Robert Manning published the equation in 1889. The roughness coefficient n is empirical, the result of decades of measurements on real channels. A smoother surface gives a lower n (faster flow); a rougher surface gives a higher n (slower flow). Typical values:

The biggest source of error in any Manning’s calculation is picking n. A 50% error in n flows through directly as a 50% error in velocity. Conservative engineering practice uses the high end of the range for capacity sizing (channel will handle the flow even if rougher than expected) and the low end for erosion protection (channel may carry faster water than expected).

Hydraulic radius R:

R = A / P, where A is cross-sectional flow area and P is wetted perimeter (the length of channel boundary in contact with the water). Shapes:

• Full circular pipe: R = D/4
• Very wide rectangular channel: R ≈ depth y
• Rectangular channel width b, depth y: R = (b·y) / (b + 2y)
• Trapezoidal channel: compute A and P from geometry

For a “very wide” channel where b » y, the wetted perimeter is dominated by the bottom width and R reduces to the depth — a useful shortcut for rivers and floodplains.

Worked example, concrete storm sewer running half full:

A 600 mm diameter concrete pipe (n = 0.013) on a 0.5% grade (S = 0.005), flowing half full. For a circular pipe at half-full depth, R = D/4 = 0.15 m.

V = (1/0.013) × 0.15^(2/3) × 0.005^(1/2) V = 76.92 × 0.2823 × 0.0707 V = 1.54 m/s

At half-full, A = πD²/8 = π·0.36/8 = 0.141 m², so Q = V·A = 1.54 × 0.141 = 0.218 m³/s (about 220 L/s).

Worked example, natural stream:

A clean natural stream with hydraulic radius 0.8 m on a gentle 0.001 slope:

V = (1/0.035) × 0.8^(2/3) × 0.001^(1/2) V = 28.57 × 0.862 × 0.0316 V = 0.78 m/s

The same channel paved with smooth concrete (n = 0.012) on the same slope would flow at 2.27 m/s — almost three times as fast. This is why urban streamlining (channelizing natural streams in concrete) causes massive downstream flood-peak increases.

Manning’s vs other methods:

• Manning’s equation is empirical but extremely well-validated for the steady-uniform-flow case.
• The Chezy equation (V = C·√(R·S)) is an older form; Manning’s is essentially Chezy with C calibrated from n.
• For laminar flow at very low velocities, neither applies; use Poiseuille for that regime.
• For rapidly varying flow (hydraulic jumps, weir overflow, sudden expansions), use the energy equation directly, not Manning’s.

Critical safety note for engineers:

A drainage ditch sized by Manning’s must also be checked for erosion velocity (typically 1.5-2 m/s for unlined earth, 4-6 m/s for concrete) and sub-critical vs supercritical regime (Froude number). Manning’s gives you velocity; it does not tell you whether that velocity is safe for the channel material.