Physics 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.

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: Product of inertia and principal moments 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 — Product of inertia and principal moments
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Production: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (3 capabilities; 189 remain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.

Product of inertiaIxy = Σ(Īxy + A (x − x̄)(y − ȳ))

Principal momentsdet([Ix, −Ixy; −Ixy, Iy] − λ I) = 0

iIx and Iy are the area-moment result. θ lies in (−90°, 90°]. Equal principal moments report θ = 0.

How to use

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

The three moments are about that part's own centroid. Axes stay parallel to x and y.

Use a negative area for a hole

Enter the hole's centroidal moments together with its negative area.

Read I1, I2, and θ

I1 is the larger principal moment. θ is its axis, counterclockwise from +x.

Example calculations

Common configurations with formula and result.

Axes already principal

Ix = 4, Iy = 1, Ixy = 0

I1 = 4, θ = 0°

I1 = 4, I2 = 1

Two unit areas on a diagonal

Zero centroidal moments at (0, 0) and (2, 2)

Ixy = 2, I1 = 4

θ = −45°

Principal 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 Ixy about the composite centroid, then the principal moments of the 2-D inertia tensor. θ is the axis of I1, counterclockwise from +x.
What it calculates: Product of inertia and principal moments of a composite plane area about its centroid.
Inputs: parts
Outputs: Ixy
I1
I2
theta_deg
Ix
Iy
x_bar
y_bar
area
Formula: Ixy = Σ(Īxy + A dx dy); principal λ from the inertia tensor
Assumptions: Calculation runs locally in the browser; values are not uploaded.
Keep units consistent with the labels on each field.
Ix and Iy are the area-moment result. θ lies in (−90°, 90°]. Equal principal moments report θ = 0.
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,Ixy,A,x,y → INVALID_INPUT
total area zero → INVALID_INPUT
Example: 0,0,0,1,0,0; 0,0,0,1,2,2 → I1 = 4, θ = −45°
Validation cases: 3 published on this page · 6/6 tests · Production surface contract 6/6 · View evidence
parts=4,1,0,2,0,0 → I1=4, I2=1, theta_deg=0
parts=0,0,0,1,0,0; 0,0,0,1,2,2 → I1=4, I2=0, theta_deg=-45
parts=0,0,0,1,0,0; 0,0,0,-1,1,0 → INVALID_INPUT
Sources: Hibbeler, Engineering Mechanics: Statics — Product of inertia and principal moments
Supports: Ixy = Īxy + A x' y' and the principal values of the plane inertia tensor
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Find the product of inertia and the principal moments of a composite plane area.

Supported and not supported

Supported: centroidal Ix, Iy, and Ixy for each part, shifted to the composite centroid, then I1, I2, and the I1 axis.

Not supported: a Mohr-circle drawing and beam deflection. Moments about a given angle, elastic Sx and Sy, and principal-axis S1 and S2, are separate pages. /calc/mechanical is not open.

Other calculators in this family: Inclined-axis moments, Principal section modulus, Area moment, Plane centroid .

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Frequently asked questions

Key distinctions behind the calculation.

Which axis is θ?

θ is the axis of the larger principal moment I1, measured counterclockwise from +x. It is folded into (−90°, 90°]. When I1 and I2 are equal, θ is 0.

Does this draw a Mohr circle?

No. It returns Ixy, I1, I2, and θ. Moments about a given angle are on the inclined-axis page. Elastic Sx and Sy, and the principal-axis moduli S1 and S2, are separate pages. A Mohr-circle figure and beam deflection stay out of scope.