Physics calculator

Hollow Circular Section Calculator

Q: Is this a solid circle?

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

Q: Does this compute bending stress?

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

Q: Is the extreme fiber the inner radius?

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

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 Engine tested · Source checked · 7/7 tests · Production surface contract 6/6 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
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 · 7/7 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 — circular area
Sources
Evidence: 3 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Production: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (6 capabilities; 189 remain CURRENT) @ 2026-09-23T14:00:05.385Z
Semantic contract: PASS

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

Enter the outer and inner radii

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

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.

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

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

Engine tested 7/7 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.0

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.
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: Circular section, Rectangular section, Section modulus .

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Frequently asked questions

Key distinctions behind the calculation.

Is this a solid circle?