HomeCalculatorsMechanicalSection PropertiesInclined-Axis Moment Calculator
Mechanical calculator

Inclined-Axis Moment Calculator

Centroidal area moments about axes rotated by a given angle. Ix, Iy, and Ixy come from the principal-moment engine. Runs locally. Not the principal values, and not a Mohr circle.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Engineering assurance
Input interpretation
Enter values to calculate.
Result
—
Assurance
Engineering
Declared partition coverage
PASS · 2/2 declared partitions (domain, invalid-domain) · Matrix
Known limitations
  • Physics L1 gap capability
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Centroidal second moments of a composite plane area about axes rotated by a given angle.
Scope
Calculation runs locally in the browser; values are not uploaded.
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Engineering assurance· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed
CVP identity
0/0 property · digest 743e1a8a564c
Legacy regression
6/6 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production STALE · overall VERIFIED_STALE
CVP status
STALE · Capability production binding: STALE · Site report: STALE · Evidence changed after the last successful production attestation. Re-attestation required.
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
1/1 golden · 3/3 CVP boundary · 3/3 invalid · 1/1 cross-interface · 6/6 CVP contract · Manifest
Sources
  • Hibbeler, Engineering Mechanics: Statics — Moments of inertia for an area about inclined axes
  • Gere and Goodno, Mechanics of Materials — Moments of inertia for an area about inclined axes
Sources
Evidence
3 legacy golden · 3 legacy boundary · legacy regression suite · 1/1 oracle-backed golden · 3/3 invalid · Artifact integrity PASS
STALE · Last attested schema matched 1.0.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · CVP STALE · attestation STALE — Production CURRENT withheld · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

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

Formulas

Core equations used by this calculator.

Ix′Ix′ = (Ix + Iy)/2 + (Ix − Iy)/2 · cos 2θ − Ixy · sin 2θ
Iy′Iy′ = (Ix + Iy)/2 − (Ix − Iy)/2 · cos 2θ + Ixy · sin 2θ
Ixy′Ixy′ = (Ix − Iy)/2 · sin 2θ + Ixy · cos 2θ
iθ is an input, counterclockwise from +x. This page does not solve for the principal angle, does not draw a Mohr circle, and does not return a section modulus.

How to use

1

Enter the parts

Each row is centroidal Ix, Iy, Ixy, A, x, y. A negative area is a hole.

2

Enter θ

Degrees, counterclockwise from +x. The primed axes are rotated by this angle.

3

Read Ix′, Iy′, and Ixy′

The unprimed moments are about axes parallel to x and y through the composite centroid.

Example calculations

Common configurations with formula and result.

ϟ

Axes already aligned

Ix = 4, Iy = 1, Ixy = 0, θ = 0°

Ix′ = Ix
Ix′ = 4, Iy′ = 1, Ixy′ = 0
ϟ

Forty-five degrees

Same area, θ = 45°

Ix′ = Iy′ = 2.5, Ixy′ = 1.5
Ixy′ = 1.5

Inclined-Axis Moment calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Ix′, Iy′, and Ixy′ about centroidal axes rotated by θ from +x. Ix, Iy, and Ixy come from the principal-moment engine.
What it calculates
Centroidal second moments of a composite plane area about axes rotated by a given angle.
Inputs
  • parts
  • theta_deg
Outputs
  • Ix_prime
  • Iy_prime
  • Ixy_prime
  • Ix
  • Iy
  • Ixy
  • theta_deg
  • x_bar
  • y_bar
  • area
Formula
Ix′ = (Ix+Iy)/2 + (Ix−Iy)/2 cos 2θ − Ixy sin 2θ
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • θ is an input, counterclockwise from +x. This page does not solve for the principal angle, does not draw a Mohr circle, and does not return a section modulus.
Units
  • θ is in degrees. Keep area, position, and second moment in one consistent unit system.
Boundary conditions
  • no parts → MISSING_REQUIRED_INPUT
  • missing θ → MISSING_REQUIRED_INPUT
  • θ is not a finite number → INVALID_INPUT
  • total area zero → INVALID_INPUT
Example
parts=4,1,0,2,0,0 theta_deg=0 → Ix′=4, Iy′=1, Ixy′=0
Validation cases

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

  • parts=4,1,0,2,0,0 theta_deg=0 → Ix_prime=4, Iy_prime=1, Ixy_prime=0
  • parts=4,1,0,2,0,0 theta_deg=45 → Ix_prime=2.5, Iy_prime=2.5, Ixy_prime=1.5
  • parts=4,1,0,2,0,0 → MISSING_REQUIRED_INPUT
Sources
  • Hibbeler, Engineering Mechanics: Statics — Moments of inertia for an area about inclined axes
    Supports: The centroidal moments transform with the double-angle formulas in θ
  • Gere and Goodno, Mechanics of Materials — Moments of inertia for an area about inclined axes
    Supports: Supports the page formula: Ix′ = (Ix + Iy)/2 + (Ix − Iy)/2 · cos 2θ − Ixy · sin 2θ
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the centroidal moments about axes rotated by a given angle.

Supported and not supported

Supported: Ix′, Iy′, and Ixy′ after the principal-moment engine finds centroidal Ix, Iy, and Ixy. θ is counterclockwise from +x.

Not supported: solving for I1, I2, and the principal angle, a Mohr-circle drawing, section modulus, bending stress, and beam deflection. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.inclined_axis_moment · tool id: inclined-axis-moment · pin 1.0.0.

{ "parts": "4,1,0,2,0,0", "theta_deg": 0 }

Share the link with parts and theta_deg. The result is not written into the link.

Other calculators in this family: Area Moment Calculator, Circular Section Calculator, Hollow Circular Section Calculator, Plane Centroid Calculator, Principal Moment Calculator, Principal Section Modulus Calculator, Rectangular Section Calculator, Section Modulus Calculator . Explore all Section Properties.

→ Area Moment Calculator Second moment of composite plane areas about the composite centroid. Supply each part's centroidal Ix and Iy. Runs locally. Not a section modulus and not a beam calculator. › → Circular Section Calculator Centroidal Ix, Iy, and elastic section moduli of a solid circle from its radius. Runs locally. › → Hollow Circular Section Calculator Centroidal Ix, Iy, and elastic section moduli of a hollow circle from the outer and inner radii. Runs locally. › → Plane Centroid Calculator Centroid of composite plane areas from each part's area and centroid. Negative area is a hole. Runs locally. Not a moment of inertia and not a beam calculator. › → Principal Moment Calculator Product of inertia and principal moments of a composite plane area. Ix and Iy reuse the area-moment engine. Runs locally. Not a section modulus and not a beam calculator. › → Principal Section Modulus Calculator Elastic section modulus about the principal axes. S1 = I1/c1 and S2 = I2/c2. I1, I2, and θ come from the principal-moment engine. Runs locally. Not Sx and Sy, and not bending stress. › → Rectangular Section Calculator Centroidal Ix, Iy, and elastic section moduli of a solid rectangle from its width and height. Runs locally. › → Section Modulus Calculator Elastic section modulus of a composite plane area about its centroid. Sx = Ix/cy and Sy = Iy/cx. Ix and Iy come from the area-moment engine. Runs locally. Not bending stress and not beam deflection. ›

Frequently asked questions

Key distinctions behind the calculation.

Is θ the principal angle?

No. You supply θ. The principal-moment page solves for the angle of I1.

Does this draw a Mohr circle?

No. It returns the three moments about the rotated axes. A Mohr-circle figure stays out of scope.

What stays constant?

Ix′ + Iy′ equals Ix + Iy. The rotation does not change that sum.