Polar Moment of Inertia Calculator

Calculator Inputs

Use consistent geometry units for section dimensions. Optional operating inputs help estimate torsional stress and twist.

Section Type

Section Dimension Unit

Solid Circle Input Type

Solid Circle Value

Outer Diameter

Inner Diameter

Mean Diameter

Wall Thickness

Applied Torque

Torque Unit

Member Length

Member Length Unit

Shear Modulus G

Modulus Unit

Formula Used

1) Solid Circular Section

Using radius: J = πr⁴ / 2

Using diameter: J = πd⁴ / 32

2) Hollow Circular Section

J = π(D_o⁴ − D_i⁴) / 32

3) Thin-Walled Circular Tube

J ≈ 2πr_m³t

Related Torsion Checks

Maximum shear stress: τ_max = Tr / J

Angle of twist: θ = TL / (JG)

Torsional section modulus: Z_p = J / r

This page calculates the area polar moment of inertia used for torsion of circular sections. The thin-walled equation is an approximation and works best when wall thickness is much smaller than mean radius.

How to Use This Calculator

Select the section type: solid circle, hollow circle, or thin-walled tube.
Choose the section unit used for geometric dimensions.
Enter the required dimensions for the chosen shape.
Optionally enter torque, member length, and shear modulus.
Press Calculate Polar Moment to show the result block above the form.
Review J, area, section modulus, radius of gyration, stress, twist, and the chart.
Use the CSV and PDF buttons to export the current results.

Example Data Table

Shape	Input	J (mm⁴)	Note
Solid Circle	d = 40 mm	251,327.412	Compact solid shaft example
Hollow Circle	Do = 60 mm, Di = 30 mm	1,192,823.461	Weight-saving tube section
Thin-Walled Tube	Dm = 80 mm, t = 4 mm	1,608,495.439	Approximation for thin tubing

Frequently Asked Questions

1) What does polar moment of inertia measure?

It measures how a cross-section resists twisting about its center. Larger values usually mean better torsional stiffness for circular members.

2) Is this the same as mass moment of inertia?

No. This calculator uses the area polar moment of inertia for section torsion. Mass moment of inertia belongs to rotational dynamics and depends on mass distribution.

3) Why are circular sections emphasized here?

Simple torsion equations directly apply to circular shafts and tubes. Noncircular sections usually require torsion constants instead of the standard circular polar formulas.

4) When should I use the thin-walled formula?

Use it when the wall thickness is small compared with the mean radius. Thick tubes should use the full hollow circle formula instead.

5) What units should I use?

Any listed unit works if dimensions stay consistent. The calculator converts internally and reports convenient SI and imperial outputs for review.

6) Why does J change so quickly with diameter?

Because the equation uses the fourth power of radius or diameter. Small size increases can produce much larger torsional resistance.

7) What do torque and shear modulus add?

They allow extra performance checks. Torque estimates maximum shear stress, while torque, length, and shear modulus together estimate twist angle.

8) Can I export the results?

Yes. After calculation, use the CSV button for tabular data or the PDF button for a clean summary report.