Physics 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

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Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary Engine tested · Source checked · 6/6 tests · Production surface contract 6/6 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
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 · 6/6 tests · Production surface contract 6/6 · Source checked · v1.0.0
Named expert review: Optional · Not performed
Sources: Hibbeler, Engineering Mechanics: Statics — Moments of inertia
Sources
Evidence: 2 golden · 4 boundary · Production surface contract 6/6 · Artifact integrity PASS
Production: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (3 capabilities; 189 remain CURRENT) @ 2026-09-23T10:40:38.140Z
Semantic contract: PASS

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

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.

Use a negative area for a hole

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

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.

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

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

Engine tested 6/6 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.0

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
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.

Other calculators in this family: Section modulus, Principal moments, Plane centroid, Planar equilibrium .

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Frequently asked questions

Key distinctions behind the calculation.

Does this build a rectangle from width and height?

No. Enter the centroidal Ix and Iy you already have. For a rectangle those are b h³/12 and h b³/12 about its own centroid.

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.