Mechanical Engineering

Polar Area Moment of a Circular Shaft

Calculate the geometric polar area moment for a solid or hollow circular shaft.

J = \frac{π}{2} (r_{o}^{4} - r_{i}^{4})

Try this formula

Calculate polar second moment of area or choose another supported quantity.

Show

Find

J = (\frac{π}{2}) \times (((r_{o})^{4} - (r_{i})^{4}))

Outer radius

Outer radius unit

Inner radius; zero for a solid shaft

Inner radius; zero for a solid shaft unit

Use decimals or scientific notation, such as 1.2e3. Calculations update as you type.

Polar second moment of area

—

Result unit

Enter the known values to see the result.

Use a concentric circular cross section with 0 ≤ r_i < r_o. J has units of length to the fourth power; it is not a mass moment of inertia. For noncircular sections, the Saint-Venant torsion constant generally differs from the polar area moment.

Unit converter Scientific calculator

Study tools

Save, share, export, or report this formula.

Source & review

Difficulty

Intermediate

Read time

5 min

Prerequisites

Basic algebra, mechanics, and consistent SI units

Review statusNeeds review

Source

raymond212 — MECH 260 — Mechanics of Materials

Reference equation: Torsion — Second Polar Moment of Area. FormuLab wrote new explanations, assumptions, applications, and numerical examples.

Definition

Understanding the core concept

Material farther from the shaft axis contributes strongly because radius enters to the fourth power. Removing material near the center can reduce weight while retaining much of a shaft’s torsional stiffness.

Variables & Units

Understanding each component

Symbol	Meaning	Units
$J$	Polar second moment of area	$m^{4}$
$r_{o}$	Outer radius	$m$
$r_{i}$	Inner radius; zero for a solid shaft	$m$

Real-World Applications

Where this formula is used in practice

Hollow shafts

Compute the section property needed for circular-shaft torsion.

Geometry comparison

Compare a solid shaft with a tubular section.

Worked Example

Step-by-step calculation with real numbers

Problem

A hollow shaft has outer radius 0.03 m and inner radius 0.02 m. Find J.

Given

r_{o} = 0.03 m

r_{i} = 0.02 m

Solution

J = \frac{π}{2} (r_{o}^{4} - r_{i}^{4})

J = \frac{π}{2} (0.0 3^{4} - 0.0 2^{4})

Final Answer

J = 1.02101761242 \times 1 0^{- 6} m^{4}

Formula Guide

Info

DifficultyIntermediate

Read Time5 min

PrerequisitesBasic algebra, mechanics, and consistent SI units

Tools

Source & review

StatusNeeds review

Last reviewedPending

Source

raymond212 — MECH 260 — Mechanics of Materials

Reference equation: Torsion — Second Polar Moment of Area. FormuLab wrote new explanations, assumptions, applications, and numerical examples.

Assumptions

• Use a concentric circular cross section with 0 ≤ r_i < r_o.
• J has units of length to the fourth power; it is not a mass moment of inertia.
• For noncircular sections, the Saint-Venant torsion constant generally differs from the polar area moment.

Languages

FormuLab is a study reference. Verify formulas, units, assumptions, and course-specific conventions before relying on them.

Browse by category

See all

Mechanical Engineering Physics Chemistry Mathematics Fluid Mechanics Electrical

Polar Area Moment of a Circular Shaft

Try this formula

Definition

Variables & Units

Real-World Applications

Hollow shafts

Geometry comparison

Worked Example

Problem

Given

Solution

Final Answer

Contents

Info

Tools

Source & review

Languages

Browse by category

Product

Company

Legal