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

Input interpretation: Enter values to calculate.
Result: —
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 · 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 for an area about inclined axes
Sources
Evidence: 3 golden · 3 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Production: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (3 capabilities; 189 remain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.

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

Enter the parts

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

Enter θ

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

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.

Axes already aligned

Ix = 4, Iy = 1, Ixy = 0, θ = 0°

Ix′ = Ix

Ix′ = 4, Iy′ = 1, Ixy′ = 0

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

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

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
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: Principal moments, Area moment .

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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.