Physics calculator

Manning Formula Calculator

Compute SI Manning discharge Q = (1/n) A R^{2/3} S^{1/2}. Runs locally. Not Q=Av, not Darcy–Weisbach, and not a weir.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary Engine tested · Source checked · 13/13 tests · Production surface contract 3/3 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
Model: SI Manning open-channel discharge and mean velocity. Invert for n or S.
Scope: SI Manning k=1
Verification: Engine tested · 13/13 tests · Production surface contract 3/3 · Source checked · v1.0.0
Named expert review: Optional · Not performed
Sources: ISO 80000-4 — Mechanics — Volume flow rate and hydraulic radius
Chow, V.T. Open-Channel Hydraulics — Manning formula
Sources
Evidence: 1 golden · 7 boundary · 5 property · Production surface contract 3/3 · Artifact integrity PASS
Production: Embedded snapshot: STALE · Last attested schema matched 1.0.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status PASS (0 stale; 163 CURRENT) @ 2026-09-17T03:09:03.307Z
Semantic contract: PASS

Formulas

Core equations used by this calculator.

DischargeQ = (1/n) A R^{2/3} S^{1/2}

Mean velocityv = (1/n) R^{2/3} S^{1/2}

iSI only (k=1). US customary 1.486 is out of scope. This page is not Q=Av, not Darcy–Weisbach, and not a weir.

How to use

Enter Manning n

n is the empirical roughness. It must be greater than zero.

Enter area A and hydraulic radius R

R may be given directly, or as A/P. Do not send R and P together.

Enter slope S

S is the friction slope (rise/run). It must be greater than zero.

Read Q and v

v = (1/n) R^{2/3} S^{1/2}. Q = A v. Invert for n or S on the other tabs.

Example calculations

Common configurations with formula and result.

Concrete channel

n=0.013, A=1 m², R=0.5 m, S=0.001

v = (1/0.013) (0.5)^{2/3} √0.001

Q = A v

From wetted perimeter

n=0.013, A=1 m², P=2 m, S=0.001

R = A/P = 0.5 m

same Q as R=0.5 m

Manning Formula calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 13/13 tests · Production surface contract 3/3
Named expert review Not performed
Calculation version 1.0.0

Review policy · Evidence

Definition: SI Manning: Q = (1/n) A R^{2/3} S^{1/2}. k = 1. Uniform open-channel flow. R = A/P.
What it calculates: SI Manning open-channel discharge and mean velocity. Invert for n or S.
Inputs: mode?
n
A
R | P
S
Q?
Outputs: Q?
v
n
A?
R
S
k
Formula: Q = (1/n) A R^{2/3} S^{1/2}
Assumptions: SI Manning k=1
Uniform open-channel flow
R = A/P
Not Q=Av identities
Not Darcy–Weisbach
Not a weir
Units: SI; A in m²; R and P in m; Q in m³/s; S dimensionless
Boundary conditions: missing n, A, R/P, or S when required → MISSING_REQUIRED_INPUT
n, A, R, P, S, or Q ≤ 0 → VALUE_MUST_BE_POSITIVE
R and P together → MIXED_INPUT_ENCODING
k given and k ≠ 1 → INVALID_MODE
weir is not a mode on this page
Example: n=0.013 A=1 R=0.5 S=0.001 → Q=(1/n) A R^{2/3} S^{1/2}
Validation cases: 3 published on this page · 13/13 tests · Production surface contract 3/3 · View evidence
n=0.013 A=1 R=0.5 S=0.001 → Q = A (1/n) R^{2/3} S^{1/2}
n=0.013 A=1 P=2 S=0.001 → R=0.5
n=0 A=1 R=0.5 S=0.001 → VALUE_MUST_BE_POSITIVE
Sources: ISO 80000-4 — Mechanics — Volume flow rate and hydraulic radius
Supports: Q is volume flow; hydraulic radius R = A/P
Chow, V.T. Open-Channel Hydraulics — Manning formula
Supports: SI form Q = (1/n) A R^{2/3} S^{1/2} for uniform flow
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Compute open-channel discharge from the SI Manning formula.

Supported and not supported

Supported — Q, v, n, or S from the SI Manning formula · API physics.fluid.manning

Not supported — Q=Av identities, Darcy–Weisbach, Colebrook, weirs, US customary k=1.486, compound channels

Agent / API notes

Capability id: physics.fluid.manning · tool id: manning-formula · pin 1.0.0.

{ "mode": "discharge", "n": 0.013, "A": 1, "R": 0.5, "S": 0.001 }

Alternate hydraulic-radius encoding: { "n": 0.013, "A": 1, "P": 2, "S": 0.001 }. Modes: discharge (default), velocity, n, slope. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, MIXED_INPUT_ENCODING, INVALID_MODE.

Calculator URL stays /calc/physics/manning-formula. There is no /calc/fluid.

Other calculators in this family: Flow rate, Darcy–Weisbach, Colebrook, Weir discharge .

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Frequently asked questions

Key distinctions behind the calculation.

Is this Q = A·v?

No. Volumetric flow from a given area and velocity is /calc/physics/flow-rate. This page finds v from Manning n, R, and S.

Is this Darcy–Weisbach?

No. Pipe head loss from a given f is /calc/physics/darcy-weisbach. Manning is uniform open-channel flow.

Is this a weir?

No. Thin-plate weir discharge is /calc/physics/weir-discharge. This page is SI Manning only.

Can I use US customary k = 1.486?

No. This page is SI only (k=1). Sending k other than 1 is rejected.

Where does this run?

Locally in the browser by default. REST and MCP call the same fluid-engine Manning wrapper.