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Trapezoidal Critical Depth Calculator

Compute trapezoid critical depth from Q, bottom width b, and side slope z. Runs locally. Not rectangular yc=(q²/g)^{1/3}, not normal depth, and not Q=Av.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Trapezoid Fr=1 Q²T = g A³ with A=(b+z y)y and T=b+2 z y; + O3 mpmath tabulated yc.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
Trapezoid Fr=1 Q²T = g A³ with A=(b+z y)y and T=b+2 z y; + O3 mpmath tabulated yc.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (depth, solve, alias, awkward, invalid-domain) · Matrix
Deferred
Not rectangular yc=(q²/g)^{1/3}, not yn, not E, not jump, and not Q=Av.
Numerical scope
O2: trapezoid Fr=1 Q²T = g A³ with A=(b+z y)y, T=b+2 z y vs a separate-module identity (≤2 ULP). Not rectangular yc=(q²/g)^{1/3}, not yn, not E, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated trapezoid-critical-depth vectors. Not rectangular yc, not yn, and not E.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Trapezoidal critical depth from Q, b, and z. Invert Q.
Scope
Trapezoid critical depth: Fr=1 with A=(b+z yc) yc and T=b+2 z yc so Q² T = g A³; SI; default g=9.80665; z>0 is run/rise
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Trapezoid Fr=1 Q²T = g A³ with A=(b+z y)y and T=b+2 z y; + O3 mpmath tabulated yc.· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-26.o2-o3
Verification revision
2026-09-26.o2-o3 · 1/1 property · digest c363ef73903f
Legacy regression
6/6 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
4/4 golden · 1/1 CVP boundary · 2/2 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 6/6 CVP contract · Manifest
Sources
  • ISO 80000-4 — Mechanics — Length, speed, acceleration
  • Chow, V.T. Open-Channel Hydraulics — Critical flow in a trapezoidal section
Sources
Evidence
1 legacy golden · legacy regression suite · 4/4 oracle-backed golden · 2/2 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Critical conditionQ² T = g A³
AreaA = (b + z y) y
Top widthT = b + 2 z y
iSI trapezoidal prismatic channel. z is horizontal/vertical. vc = Q/A = √(g A/T) is reported. Fr=1 is the definition of critical flow, not a Froude-number wrap. Not rectangular yc=(q²/g)^{1/3}, not yn, not E, not Manning, not Q=Av.

How to use

1

Enter Q, b, and z

Discharge in m³/s, bottom width in m, and side slope z (run/rise). z=0 is the rectangular page.

2

Read yc

Solve Q² T = g A³ for y. Invert Q on the Q tab when yc is known.

3

Optional g

Gravity defaults to 9.80665 m/s².

Example calculations

Common configurations with formula and result.

ϟ

Q=2 m³/s, b=2 m, z=1

Default g=9.80665 m/s²

Q² T = g A³
yc ≈ 0.43298 m
ϟ

Invert Q

yc ≈ 0.43298 m, b=2 m, z=1

Q = √(g A³ / T)
Q ≈ 2 m³/s

Trapezoidal Critical Depth calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Trapezoid critical depth: Fr=1 with A=(b+z y)y and T=b+2 z y so Q² T = g A³. Default g=9.80665 m/s². z is run/rise, z>0. Rectangular yc=(q²/g)^{1/3} is a different page.
What it calculates
Trapezoidal critical depth from Q, b, and z. Invert Q.
Inputs
  • mode?
  • Q|yc
  • b
  • z
  • g?
Outputs
  • yc
  • Q
  • b
  • z
  • A
  • T
  • D
  • vc
  • Fr
  • g
  • mode
  • model
Formula
Q² T = g A³
Assumptions
  • Trapezoid critical depth: Fr=1 with A=(b+z yc) yc and T=b+2 z yc so Q² T = g A³; SI; default g=9.80665; z>0 is run/rise
  • Critical speed vc = Q/A = √(g A/T) is reported, not an input; Fr=1 is the definition, not a Froude wrap
  • Not rectangular yc=(q²/g)^{1/3}; not normal-depth yn; not specific energy E; not Manning n; not Q=Av
Units
  • SI; yc in m; Q in m³/s; b in m; z dimensionless; g in m/s²
Boundary conditions
  • missing Q, b, z, or yc when required → MISSING_REQUIRED_INPUT
  • Q, b, z, yc, or g ≤ 0 → VALUE_MUST_BE_POSITIVE
  • z=0, q, n/S/yn, E, Fr/L, A/v, y1/M/Sc, B, theta → MIXED_INPUT_ENCODING
  • z>10 or yc/b outside [0.01, 20] → VALUE_OUT_OF_RANGE
  • unit_discharge, energy, or yn as mode → MIXED_INPUT_ENCODING or INVALID_MODE
Example
Q=2 b=2 z=1 → yc≈0.43298 m
Validation cases

4 published on this page · 6/6 tests · Production surface contract 6/6 · View evidence

  • Q=2 b=2 z=1 → yc ≈ 0.43298 m, Fr=1
  • mode=discharge yc≈0.43298 b=2 z=1 → Q ≈ 2 m³/s
  • Q=2 b=2 z=0 → MIXED_INPUT_ENCODING
  • Q=2 b=2 z=1 q=1 → MIXED_INPUT_ENCODING
Sources
  • ISO 80000-4 — Mechanics — Length, speed, acceleration
    Supports: yc is a length; Q is volume flow; g is acceleration
  • Chow, V.T. Open-Channel Hydraulics — Critical flow in a trapezoidal section
    Supports: Fr=1 is Q² T = g A³. Rectangular yc=(q²/g)^{1/3} is the z=0 special case on a different page.
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute trapezoidal critical depth from discharge Q, bottom width b, and side slope z.

Supported and not supported

Supported — yc from Q, b, z · invert Q · API physics.fluid.trapezoid_critical_depth

Not supported — rectangular yc=(q²/g)^{1/3}; normal-depth yn; specific energy E; Manning n; Q=Av; z=0

Agent / API notes

Capability id: physics.fluid.trapezoid_critical_depth · tool id: trapezoidal-critical-depth · pin 1.0.0.

{ "Q": 2, "b": 2, "z": 1 }

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

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

Other calculators in this family: Alternate Depth Calculator, Bernoulli Equation Calculator, Broad-Crested Weir Calculator, Cavitation Number Calculator, Chezy Formula Calculator, Colebrook Friction Factor Calculator, Concentric Reducer Calculator, Critical Depth Calculator . Explore all Fluid Mechanics.

→ Alternate Depth Calculator Compute both rectangular depths at the same specific energy from E, Q, and width b. Runs locally. Not specific energy from y+v; not specific force M; not hydraulic jump y2; not critical depth yc as the product. › → Bernoulli Equation Calculator Incompressible Bernoulli along a streamline: solve P2, v2, or h2. Runs locally in your browser. › → Broad-Crested Weir Calculator Compute rectangular broad-crested weir discharge Q = C_d (2/3)^{3/2} b √g H^{3/2}. Default C_d=0.85. Runs locally. Not a thin-plate weir, not Manning, not Q=Av, and not an orifice. › → Cavitation Number Calculator Compute the cavitation number σ = (p − p_v) / (½ ρ v²). Runs locally. Not NPSH, not Thoma NPSH/H, not Weber, and not Euler Eu=Δp/(ρ v²). › → Chezy Formula Calculator Compute open-channel mean velocity V = C √(R S). Runs locally. Not Manning n, not Q=Av, not Hazen–Williams, and not Darcy. › → Colebrook Friction Factor Calculator Solve the Colebrook–White equation for Darcy friction factor f from Re and ε/D. Turbulent pipes Re≥4000. Runs locally. Not laminar 64/Re and not a Moody-chart lookup. › → Concentric Reducer Calculator Compute concentric-reducer loss h = K v₂² / (2g) with K = 0.5 (1 − A₂/A₁) referenced to the downstream speed. Runs locally. Not sudden expansion, not gradual contraction α, not vena-contracta C_c, not catalog K, and not Q=Av. › → Critical Depth Calculator Compute rectangular critical depth yc = (q²/g)^{1/3}. Runs locally. Not a Froude wrap, not specific energy, not a hydraulic jump, and not Q=Av. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this rectangular critical depth?

No. yc = (q²/g)^{1/3} is /calc/physics/critical-depth. This page needs z>0. Sending z to the rectangular page, or q to this page, is MIXED_INPUT_ENCODING. z=0 is MIXED_INPUT_ENCODING.

Is this normal depth?

No. Trapezoid yn from Manning is /calc/physics/trapezoidal-normal-depth. Rectangular yn is /calc/physics/normal-depth. Sending n, S, or yn is MIXED_INPUT_ENCODING.

Is this the Froude number?

No. Fr = v / √(g L) is /calc/physics/froude-number. This page returns yc at Fr=1. Sending Fr or L is MIXED_INPUT_ENCODING.

Is this Q = A·v?

No. Volumetric flow from area and velocity is /calc/physics/flow-rate. Sending A or v is MIXED_INPUT_ENCODING.

Where does this run?

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