HomeCalculatorsMechanicalSection PropertiesHollow Circular Section Calculator
Mechanical calculator

Hollow Circular Section Calculator

Centroidal Ix, Iy, and elastic section moduli of a hollow circle from the outer and inner radii. Runs locally.

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 area moments, radii of gyration, and elastic section moduli of a hollow circle.
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 68f8a9e7a37a
Legacy regression
7/7 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 · 4/4 invalid · 1/1 cross-interface · 6/6 CVP contract · Manifest
Sources
  • Hibbeler, Engineering Mechanics: Statics — Moments of inertia — circular area
  • Gere and Goodno, Mechanics of Materials — Moments of inertia — circular area
Sources
Evidence
3 legacy golden · 4 legacy boundary · legacy regression suite · 1/1 oracle-backed golden · 4/4 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.

About a centroidal diameterIx = Iy = π(R⁴−r⁴)/4
Elastic section modulusSx = Sy = Ix/R
ikx = ky = √(R²+r²)/2. R is the outer radius and r is the inner radius. A solid circle is the circular-section page.

How to use

1

Enter the outer and inner radii

R and r are positive, and r is less than R. They set the length unit.

2

Read the centroidal moments and the moduli

Ix and Iy are equal about any centroidal diameter. Sx and Sy divide those moments by the outer radius.

Example calculations

Common configurations with formula and result.

ϟ

Outer 2, inner 1

R = 2, r = 1

Ix = 15π/4, Sx = 15π/8, kx = √5/2
Ix = 11.780972, Sx = 5.890486, kx = 1.118034
ϟ

Outer 4, inner 2

R = 4, r = 2

Ix = 60π, Sx = 15π, kx = √5
Ix = 188.495559, Sx = 47.123890, kx = 2.236068

Hollow Circular Section calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
For a hollow circle, centroidal Ix = Iy = π(R⁴−r⁴)/4. The elastic moduli are Sx = Sy = Ix/R, and the radius of gyration is √(R²+r²)/2.
What it calculates
Centroidal area moments, radii of gyration, and elastic section moduli of a hollow circle.
Inputs
  • R
  • r
Outputs
  • Ix
  • Iy
  • Sx
  • Sy
  • area
  • kx
  • ky
Formula
Ix = Iy = π(R⁴−r⁴)/4, Sx = Sy = Ix/R, kx = ky = √(R²+r²)/2
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • kx = ky = √(R²+r²)/2. R is the outer radius and r is the inner radius. A solid circle is the circular-section page.
Units
  • R and r set the length unit. Area is that unit squared. Ix and Iy are that unit to the fourth. Sx and Sy are that unit cubed. kx and ky are lengths.
Boundary conditions
  • missing R or r → MISSING_REQUIRED_INPUT
  • R ≤ 0 or r ≤ 0 → VALUE_MUST_BE_POSITIVE
  • r ≥ R → INVALID_INPUT
Example
R=2 r=1 → area=3π, Ix=15π/4, Sx=15π/8, kx=√5/2
Validation cases

4 published on this page · 7/7 tests · Production surface contract 6/6 · View evidence

  • R=2 r=1 → area=3π, Ix=Iy=15π/4, Sx=Sy=15π/8, kx=ky=√5/2
  • R=4 r=2 → area=12π, Ix=Iy=60π, Sx=Sy=15π, kx=ky=√5
  • r=0 → VALUE_MUST_BE_POSITIVE
  • r=R → INVALID_INPUT
Sources
  • Hibbeler, Engineering Mechanics: Statics — Moments of inertia — circular area
    Supports: The centroidal moment of a hollow circle about a diameter is π(R⁴−r⁴)/4.
  • Gere and Goodno, Mechanics of Materials — Moments of inertia — circular area
    Supports: Supports the page formula: Ix = Iy = π(R⁴−r⁴)/4
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the centroidal moments and elastic section moduli of a hollow circle.

Supported and not supported

Supported: outer radius R and inner radius r, with r < R. Ix = Iy = π(R⁴−r⁴)/4 is passed to the area-moment engine. Sx and Sy come from the section-modulus engine at the outer fiber R. The radius of gyration is √(R²+r²)/2.

Not supported: a solid circle, a rectangle, bending stress, and the plastic modulus. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.hollow_circular_section · tool id: hollow-circular-section · pin 1.0.0.

{ "R": 2, "r": 1 }

Share the link with R and r. The result is not written into the link.

Other calculators in this family: Area Moment Calculator, 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.

→ 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. › → 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.

Is this a solid circle?

A solid circle from one radius is the circular-section page. This page keeps a hole.

Does this compute bending stress?

No. It returns the elastic section modulus. Bending stress, the plastic modulus, and material allowables are out of scope.

Is the extreme fiber the inner radius?

No. Sx and Sy divide the centroidal moment by the outer radius.