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Manning's Equation Calculator — With FE & PE Exam Worked Examples
Solve Manning's equation for discharge, normal depth, slope, or roughness in rectangular, trapezoidal, and circular channels — in US or SI units. The calculator is pre-loaded with a real exam-style problem; every worked example below reproduces in the tool above.
Switching converts every value into the new system (it doesn't just relabel). Your choice is remembered on this device.
For study and checking only. These calculators run entirely in your browser — nothing is sent anywhere. They are meant to verify your hand calculations while you study; on the exam you will work by hand with the NCEES reference handbook.
Quick answer: Manning's equation
US units: V = (1.486/n) · R2/3 · S1/2, Q = V·A | SI units: V = (1.0/n) · R2/3 · S1/2, Q = V·A
The constant is 1.486 in US units and 1.0 in SI — never 1.49. Mixing the two is the single most common way this equation is missed on exams: using 1.486 with metres, or 1.0 with feet, shifts the answer by roughly 49%.
Manning's equation in all four rearranged forms
| V | mean velocity, ft/s (m/s) — V = (k/n)·R2/3·S1/2 |
| Q | discharge, cfs (m³/s) — Q = V·A |
| S | energy (bed) slope, ft/ft — S = [ Q·n / (k·A·R2/3) ]² |
| n | Manning's roughness — n = k·A·R2/3·S1/2 / Q |
| y | normal depth — solved by iteration: find y with Q(y) = Qtarget |
| R = A/P | hydraulic radius — area divided by wetted perimeter (the free surface is never wetted) |
The constant k = 1.486 (US) / 1.0 (SI). Gravity enters only through the Froude number used for the flow-regime check: Fr = V/√(g·Dh), with g = 32.2 ft/s² (9.81 m/s²) and hydraulic depth Dh = A/T.
How to use it — and how to sanity-check the answer
- Pick what you're solving for. Most exam problems ask for Q, but "find the normal depth" and "find the required slope" are standard variants — the tool handles all four.
- Match the shape. Trapezoidal covers most earth and lined channels; rectangular is the z = 0 special case; circular covers sewers and culverts flowing partially full.
- Enter the slope as a ratio (0.001, not 0.1) — or flip the field to % entry. A slope of "0.1" typed as a ratio means a 10% grade, and the answer will be absurdly large.
- Check V and R before trusting Q. Most exam channels run V ≈ 1–15 ft/s (0.3–4.6 m/s) and R ≈ 0.5–10 ft (0.15–3 m). A result far outside those bands almost always means a unit slip — usually the 1.486/1.0 constant or inches fed to a feet formula.
Worked example 1 (PE pace): trapezoidal normal flow
Hydraulics — Open Channel. A trapezoidal channel has bottom width b = 3 m, side slopes z = 2 (H:1V), and carries normal depth y = 1.5 m on a bed slope S = 0.001 with Manning's n = 0.013. Find the discharge. Switch the calculator to SI above and enter these values — it reproduces every number below.
- A = (b + z·y)·y = (3 + 2×1.5)×1.5 = 9.00 m²
- P = b + 2y√(1+z²) = 3 + 2(1.5)√5 = 3 + 6.708 = 9.71 m
- R = A/P = 9.000/9.708 = 0.93 m; R2/3 = 0.9508
- V = (1.0/0.013)(0.9508)(0.001)1/2 = 76.923 × 0.9508 × 0.031623 = 2.31 m/s
- Q = V·A = 2.3127 × 9.000 = 20.81 m³/s
- Regime check: top width T = b + 2zy = 9.00 m, hydraulic depth Dh = A/T = 1.00 m, Fr = 2.313/√(9.81×1.000) = 0.738 — subcritical.
The same problem in US units (b = 9.843 ft, y = 4.921 ft, n = 0.013, S = 0.001 — the calculator's default load) gives A = 96.88 ft², R = 3.04 ft, V = 7.59 ft/s, Q = 735.1 cfs, Fr = 0.738.
Worked example 2 (FE pace, ~3 min): rectangular channel
A rectangular concrete channel is 10 ft wide with normal depth 4 ft, bed slope 0.002, Manning's n = 0.015. Find the discharge.
- A = 10 × 4 = 40.00 ft²; P = 10 + 2(4) = 18.00 ft; R = 40/18 = 2.22 ft
- V = (1.486/0.015)(2.222)2/3(0.002)1/2 = 99.067 × 1.703 × 0.044721 = 7.54 ft/s
- Q = 7.5446 × 40 = 301.8 cfs
- Fr = 7.545/√(32.2 × 4.000) = 0.665 — subcritical.
Set the calculator to rectangular, solve-for Q, and enter these values to verify.
Worked example 3: rectangular channel (SI)
A rectangular concrete channel is 2.0 m wide with normal depth 0.80 m, bed slope 0.001, Manning's n = 0.013. Find the discharge.
- A = 2.0 × 0.80 = 1.60 m²
- P = 2.0 + 2(0.80) = 3.60 m → R = A/P = 0.444 m
- V = (1/0.013)(0.444)2/3(0.001)1/2 = 1.42 m/s
- Q = AV ≈ 2.27 m³/s
Exam tip: the wetted perimeter of a rectangular channel excludes the free surface — the most common way this question is missed.
Free formula resources
Grab the free formula resources — every FE Civil equation in one searchable index, including all four rearranged forms of Manning's.
Manning's n — reference table
Values are ranges from standard FHWA/Chow references. The calculator's presets insert the range midpoint (labelled as such) — for graded work, pick the value your reference specifies.
| Channel / lining | n range | Typical use |
|---|---|---|
| Finished concrete | 0.011–0.015 | lined channels, culverts |
| Unfinished concrete | 0.014–0.017 | shotcrete, rough formed |
| Smooth asphalt | 0.013–0.016 | paved channels |
| Corrugated metal pipe | 0.021–0.026 | CMP culverts (varies by corrugation) |
| Cast iron | 0.011–0.015 | older pressure-style pipe |
| PVC / HDPE smooth plastic | 0.009–0.011 | smoothest common lining |
| Earth, clean & straight | 0.018–0.025 | maintained earth channel |
| Earth, gravelly | 0.025–0.030 | unmaintained earth |
| Natural stream, clean winding | 0.033–0.045 | natural channels |
| Floodplain, light brush | 0.035–0.060 | overbank flow |
| Floodplain, heavy brush/timber | 0.060–0.120 | wooded overbank |
Common exam traps
- R is not depth. Hydraulic radius is A/P — area over wetted perimeter. The only time R ≈ depth is the wide-channel approximation, and the exam will say so.
- Wetted perimeter excludes the free surface. For a rectangular channel P = b + 2y, not 2b + 2y. For a circular pipe flowing half full, P = πD/2 — and note the anchor: a half-full and a completely full circular channel have the same R = D/4.
- The unit constant. k = 1.486 (US) / 1.0 (SI) — never 1.49. If your answer is ~49% off a choice, you mixed the constants.
- Maximum circular-pipe discharge is at y/D ≈ 0.938, not full. A pipe at 100% full carries slightly less than at ~94% full. The solve-for-depth mode above restricts the search to the unique-solution branch and errors out cleanly if the target exceeds it.
- Normal depth vs critical depth. Manning's gives normal depth (uniform flow). If the problem asks for critical depth, that's a specific-energy problem — different equation entirely.
- Slope as a ratio. S = 0.001 means 0.1%. Typing 0.1 when you mean 0.1% inflates V by √1000 ≈ 32×.
Manning's equation — FAQ
Is the hydraulic radius ever equal to the flow depth?
Only in the wide-channel approximation: when a channel is very wide relative to its depth, P ≈ b and A ≈ b·y, so R ≈ y. For ordinary channels the exam expects the real R = A/P — and graders love problems where the wide-channel shortcut is wrong.
Which constant — 1.486 or 1.49?
1.486 in US units, 1.0 in SI. The NCEES reference handbook prints 1.486; "1.49" is a rounded memory aid that shifts answers by ~0.3% — harmless alone, but fatal combined with a second slip. Use 1.486 exactly.
Can Manning's equation be used for pipes?
Yes — for pipes flowing partially full under gravity (storm drains, sewers, culverts), Manning's with circular geometry is the standard exam approach. For pressurized full pipes, the exam expects Hazen–Williams or Darcy–Weisbach instead — see the Hazen–Williams calculator.
Why is my answer off by ~49%?
You almost certainly mixed the unit constants — used 1.486 with metres or 1.0 with feet. The ratio 1.486/1.0 ≈ 1.49 is the fingerprint. Re-check the constant before anything else.
What n value should I pick for a corrugated metal pipe?
The standard range is 0.021–0.026 depending on corrugation size. The calculator preset inserts 0.0235 (the range midpoint, labelled as such); on the exam, use whatever the problem statement or the reference handbook table gives you.
Why does a full pipe carry less than a nearly-full pipe?
Discharge is V·A, and near full the wetted perimeter grows faster than the area — friction wins. Maximum conveyance for a circular section is at about 94% of the diameter, which is why the depth-solver here caps its search there.
Keep building exam speed
Open-channel flow is 7–11 of the 80 questions on the PE Civil Water Resources & Environmental exam. The PE WRE Flagship ($119) drills every one of those question types — soft launch, join the waitlist from the contact page. For a full rehearsal under timed conditions, the 110-question FE Civil practice exam ($79) is a 5-hour-20-minute run with detailed solutions.
Last reviewed: 2026-10-03. Formulas follow the NCEES FE Reference Handbook conventions; always confirm against the current handbook.