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Trapezoidal Normal Depth Calculator

Compute trapezoid uniform-flow yn from SI Manning with A=(b+z y)y and P=b+2 y √(1+z²). Runs locally. Not rectangular yn, not trapezoid yc, not Manning A,R, not Chezy, not Q=Av.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Source checked · 5/5 tests · Production surface contract 1/1 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
Trapezoidal uniform-flow normal depth from Q, n, S, b, and z. Invert Q.
Scope
Trapezoid uniform flow: Q=(1/n) A R^{2/3} S^{1/2}; A=(b+z yn) yn; P=b+2 yn √(1+z²); SI k=1; z>0 is run/rise; yn is unknown in depth mode
Verification
Engine tested · 5/5 tests · Production surface contract 1/1 · 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 — Normal depth in a trapezoidal section
Sources
Evidence
5 property · Production surface contract 1/1 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status PASS (0 stale; 164 CURRENT) @ 2026-09-18T21:41:56.329Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

ManningQ = (1/n) A R^{2/3} S^{1/2}
AreaA = (b + z y) y
Wetted perimeterP = b + 2 y √(1+z²)
iSI k=1. z is run/rise, z>0. Geometry is bottom width b and side slope z, not A and R as inputs. yn/b∈[0.01, 10], n∈[0.008, 0.15], S∈[1e-6, 0.1], z∈(0, 10]. z=0 is the rectangular page. Not US k=1.486. Do not send query v — that is calculation_version. Mean speed is result v.

How to use

1

Enter Q, n, S, b, and z

Discharge, Manning n, bed slope, bottom width, and side slope z (run/rise). Invert on the Q tab when yn is known. z=0 is the rectangular page.

2

Read yn

yn is the uniform-flow depth. A, P, R, and v=Q/A are reported.

Example calculations

Common configurations with formula and result.

ϟ

b=2 m, z=1, n=0.03, S=0.001, Q≈2.30256 m³/s

Trapezoidal channel

A=(b+z yn) yn, P=b+2 yn √(1+z²), Q=(1/n) A R^{2/3} S^{1/2}
yn = 1.0000 m
ϟ

Invert Q

yn = 1 m, same n, S, b, z

Q from trapezoid Manning
Q ≈ 2.30256 m³/s

Trapezoidal Normal Depth calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Trapezoid uniform flow: Q = (1/n) A R^{2/3} S^{1/2} with A = (b + z y) y and P = b + 2 y √(1+z²). This page solves yn. Rectangular yn and Manning with given A and R are different pages.
What it calculates
Trapezoidal uniform-flow normal depth from Q, n, S, b, and z. Invert Q.
Inputs
  • mode?
  • b
  • z
  • n
  • S
  • Q|yn
Outputs
  • yn
  • Q
  • b
  • z
  • n
  • S
  • A
  • P
  • R
  • v
  • mode
  • model
Formula
Q = (1/n) A R^{2/3} S^{1/2}; A = (b + z yn) yn; P = b + 2 yn √(1+z²)
Assumptions
  • Trapezoid uniform flow: Q=(1/n) A R^{2/3} S^{1/2}; A=(b+z yn) yn; P=b+2 yn √(1+z²); SI k=1; z>0 is run/rise; yn is unknown in depth mode
  • yn/b∈[0.01, 10]; n∈[0.008, 0.15]; S∈[1e-6, 0.1]; z∈(0, 10]; geometry is b and z, not A and R
  • Not rectangular yn; not trapezoid yc; not Manning Q with given A,R; not Chezy C; not Q=Av
Units
  • SI; yn, b in m; Q in m³/s; S dimensionless; z dimensionless; n SI
Boundary conditions
  • missing b, z, n, S, Q, or yn when required → MISSING_REQUIRED_INPUT
  • yn/b, n, S, or z outside envelope → VALUE_OUT_OF_RANGE
  • z=0, A/R, yc/q, E, Chezy C, v, B, theta, g/Fr → MIXED_INPUT_ENCODING
  • unknown mode → INVALID_MODE
Example
Q=2.30256 b=2 z=1 n=0.03 S=0.001 → yn=1.0000
Validation cases

4 published on this page · 5/5 tests · Production surface contract 1/1 · View evidence

  • Q=2.30256 b=2 z=1 n=0.03 S=0.001 → yn ≈ 1.0000
  • mode=discharge yn=1 b=2 z=1 n=0.03 S=0.001 → Q ≈ 2.30256
  • Q=2.30256 b=2 z=0 n=0.03 S=0.001 → MIXED_INPUT_ENCODING
  • Q=2.30256 b=2 z=1 n=0.03 S=0.001 A=3 R=0.62 → MIXED_INPUT_ENCODING
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 — Normal depth in a trapezoidal section
    Supports: Uniform-flow yn from SI Manning with trapezoid A=(b+z y)y and P=b+2 y √(1+z²). Rectangular A=b yn is a different page. Not Manning with given A,R as a product page, not yc, not Chezy C
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute trapezoidal uniform-flow normal depth yn from Q, n, S, bottom width b, and side slope z.

Supported and not supported

Supported — yn from Q, n, S, b, z · invert Q · API physics.fluid.trapezoid_normal_depth

Not supported — rectangular yn; trapezoid yc; Manning A,R; Chezy C; Q=Av; z=0; US k=1.486

Agent / API notes

Capability id: physics.fluid.trapezoid_normal_depth · tool id: trapezoidal-normal-depth · pin 1.0.0.

{ "Q": 2.30256, "b": 2, "z": 1, "n": 0.03, "S": 0.001 }

REST and MCP call the same trapezoidNormalDepth wrapper as the page.

Calculator URL stays /calc/physics/trapezoidal-normal-depth. There is no /calc/fluid.

Other calculators in this family: Normal depth, Trapezoidal critical depth, Trapezoidal specific energy, Trapezoidal alternate depth, Manning formula, Chezy formula .

yn Normal depth › yc Trapezoidal critical depth › E Trapezoidal specific energy › y Trapezoidal alternate depth › Q Manning formula › C Chezy formula ›
}

Frequently asked questions

Key distinctions behind the calculation.

Is this rectangular normal depth?

No. Rectangular yn from A=b yn is /calc/physics/normal-depth. This page needs z>0. Sending z to the rectangular page, or z=0 here, is MIXED_INPUT_ENCODING.

Is this the Manning formula page?

No. /calc/physics/manning-formula takes A and R as inputs. This page takes b and z and solves yn. Sending A or R is MIXED_INPUT_ENCODING.

Is this trapezoidal critical depth?

No. Trapezoid Fr=1 from Q, b, z is /calc/physics/trapezoidal-critical-depth. Sending yc, q, or g is MIXED_INPUT_ENCODING.

Is this Chezy?

No. V = C √(R S) is /calc/physics/chezy-formula. Sending Chezy C is MIXED_INPUT_ENCODING.

Is this Q = A·v?

No. Volumetric flow from a given area and velocity is /calc/physics/flow-rate. Sending v is MIXED_INPUT_ENCODING. Result v = Q/A is reported.

Where does this run?

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