HomeCalculatorsMechanicalSection PropertiesArea Moment Calculator
Mechanical calculator

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.

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
Second moments and radii of gyration of a composite plane area about its centroid.
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 6fb9f8d9f563
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 · 4/4 CVP boundary · 3/3 invalid · 1/1 cross-interface · 6/6 CVP contract · Manifest
Sources
  • Hibbeler, Engineering Mechanics: Statics — Moments of inertia
  • Gere and Goodno, Mechanics of Materials — Moments of inertia
Sources
Evidence
2 legacy golden · 4 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.

Parallel axisIx = Σ(Īx + A (y − ȳ)²)
Radius of gyrationkx = √(Ix / A), ky = √(Iy / A)
iThe centroid is the plane-centroid result. A rectangle's own Īx = b h³/12 is entered by the user. This page does not compute that rectangle, a product of inertia, or a section modulus.

How to use

1

Enter each part as Ix, Iy, A, x, y

Ix and Iy are about that part's own centroid. The axes stay parallel to x and y.

2

Use a negative area for a hole

Subtract the hole's centroidal moments by entering them with the hole's negative area.

3

Read Ix, Iy, and the radii of gyration

Both moments are about the composite centroid. kx and ky are omitted when I/A is negative.

Example calculations

Common configurations with formula and result.

ϟ

Two unit areas

Each part has Īx = 1 and area 1, at y = 0 and y = 2

ȳ = 1, Ix = 1 + 1·1² + 1 + 1·1²
Ix = 4
ϟ

One part at the origin

Īx = 16, Īy = 4, area 4

kx = √(16/4)
kx = 2, ky = 1

Area Moment calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Composite second moments about the composite centroid. Ix = Σ(Īx + A (y − ȳ)²) and Iy = Σ(Īy + A (x − x̄)²). Īx and Īy are centroidal and parallel to the global axes.
What it calculates
Second moments and radii of gyration of a composite plane area about its centroid.
Inputs
  • parts
Outputs
  • Ix
  • Iy
  • kx
  • ky
  • x_bar
  • y_bar
  • area
Formula
Ix = Σ(Īx + A (y−ȳ)²), Iy = Σ(Īy + A (x−x̄)²)
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • The centroid is the plane-centroid result. A rectangle's own Īx = b h³/12 is entered by the user. This page does not compute that rectangle, a product of inertia, or a section modulus.
Units
  • Keep area, position, and second moment in one consistent unit system. A negative area is a hole.
Boundary conditions
  • no parts → MISSING_REQUIRED_INPUT
  • a row that is not Ix,Iy,A,x,y → INVALID_INPUT
  • total area zero → INVALID_INPUT
Example
1,0,1,0,0; 1,0,1,0,2 → Ix = 4, ȳ = 1
Validation cases

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

  • parts=1,0,1,0,0; 1,0,1,0,2 → Ix=4, Iy=0, y_bar=1
  • parts=16,4,4,0,0 → kx=2, ky=1
  • parts=0,0,1,0,0; 0,0,-1,1,0 → INVALID_INPUT
Sources
  • Hibbeler, Engineering Mechanics: Statics — Moments of inertia
    Supports: Parallel-axis theorem I = Ī + A d² for a composite area
  • Gere and Goodno, Mechanics of Materials — Moments of inertia
    Supports: Supports the page formula: Ix = Σ(Īx + A (y − ȳ)²)
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the second moment of a composite plane area about its centroid.

Supported and not supported

Supported: centroidal Ix and Iy for each part, shifted to the composite centroid, plus the radii of gyration.

Not supported: building a rectangle from width and height, product of inertia, and principal axes. Elastic section modulus is a separate page. Beam deflection stays closed. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.area_moment · tool id: area-moment · pin 1.0.0.

{ "parts": "1,0,1,0,0; 1,0,1,0,2" }

Each part is Ix,Iy,A,x,y. Share the link with parts. The result is not written into the link.

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

→ 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. › → 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. › → 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.

Does this build a rectangle from width and height?

A solid rectangle from width and height is the rectangular-section page. This page takes centroidal Ix and Iy you already have, including a composite.

Does this return a section modulus?

No. Elastic section modulus is on the section-modulus page. This page returns Ix, Iy, and the radii of gyration about the composite centroid.