Physics calculator

Rectangular Section Calculator

Centroidal Ix, Iy, and elastic section moduli of a solid rectangle from its width and height. Runs locally.

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 area moments, radii of gyration, and elastic section moduli of a solid rectangle.
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 — rectangular area
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Production: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (6 capabilities; 189 remain CURRENT) @ 2026-09-23T14:00:05.385Z
Semantic contract: PASS

Formulas

Core equations used by this calculator.

About the centroidIx = b h³/12, Iy = h b³/12

Elastic section modulusSx = b h²/6, Sy = h b²/6

ib is parallel to x. h is parallel to y. A composite section is the area-moment page. A circle is its own page.

How to use

Enter the width and the height

Both dimensions are positive and use the same length unit.

Read the centroidal moments and the moduli

Ix and Iy are about axes through the center, parallel to the sides. Sx and Sy divide those moments by the distance to the extreme fiber.

Example calculations

Common configurations with formula and result.

Tall rectangle

b = 2, h = 6

Ix = 36, Sx = 12

Ix = 36, Iy = 4, Sx = 12, Sy = 4

Square

b = 1, h = 1

Ix = 1/12, Sx = 1/6

Ix = Iy = 1/12, Sx = Sy = 1/6

Wide rectangle

b = 6, h = 2

Ix = 4, Iy = 36

Ix = 4, Iy = 36, Sx = 4, Sy = 12

Rectangular Section 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: For a solid rectangle, centroidal Ix = b h³/12 and Iy = h b³/12. The elastic moduli are Sx = b h²/6 and Sy = h b²/6.
What it calculates: Centroidal area moments, radii of gyration, and elastic section moduli of a solid rectangle.
Inputs: b
h
Outputs: Ix
Iy
Sx
Sy
area
kx
ky
Formula: Ix = b h³/12, Iy = h b³/12, Sx = b h²/6, Sy = h b²/6
Assumptions: Calculation runs locally in the browser; values are not uploaded.
Keep units consistent with the labels on each field.
b is parallel to x. h is parallel to y. A composite section is the area-moment page. A circle is its own page.
Units: b and h share one 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 b or h → MISSING_REQUIRED_INPUT
b ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
Example: b=2 h=6 → area=12, Ix=36, Iy=4, Sx=12, Sy=4
Validation cases: 4 published on this page · 6/6 tests · Production surface contract 6/6 · View evidence
b=2 h=6 → area=12, Ix=36, Iy=4, Sx=12, Sy=4
b=1 h=1 → Ix=Iy=1/12, Sx=Sy=1/6
b=6 h=2 → Ix=4, Iy=36, Sx=4, Sy=12
b=0 h=6 → VALUE_MUST_BE_POSITIVE
Sources: Hibbeler, Engineering Mechanics: Statics — Moments of inertia — rectangular area
Supports: The centroidal moments of a rectangle are b h³/12 and h b³/12.
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Find the centroidal moments and elastic section moduli of a solid rectangle.

Supported and not supported

Supported: width b parallel to x and height h parallel to y. Ix = b h³/12 and Iy = h b³/12 are passed to the area-moment engine. Sx and Sy come from the section-modulus engine at the extreme fibers h/2 and b/2.

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

Agent / API notes

Capability id: mechanical.statics.rectangular_section · tool id: rectangular-section · pin 1.0.0.

{ "b": 2, "h": 6 }

Share the link with b and h. The result is not written into the link.

Other calculators in this family: Circular section, Area moment, Section modulus, Plane centroid .

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Frequently asked questions

Key distinctions behind the calculation.

Is this a composite section?

No. A composite of several areas, including a hole, is the area-moment page. This page is one solid rectangle.

Does this compute bending stress?

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

Does this cover a circle?

A solid circle from its radius is the circular-section page.