Calculator Inputs
This tool computes the geometric polar moment of area for common shapes.
Example Data Table
| Shape | Inputs | Formula | J Result |
|---|---|---|---|
| Solid circle | D = 40 mm | J = πD⁴ / 32 | 251,327.412287 mm^4 |
| Hollow circle | Do = 60 mm, Di = 30 mm | J = π(Do⁴ − Di⁴) / 32 | 1.192823e+6 mm^4 |
| Rectangle | b = 40 mm, h = 80 mm | J = bh(b² + h²) / 12 | 2.133333e+6 mm^4 |
| Ellipse | a = 30 mm, b = 15 mm | J = πab(a² + b²) / 4 | 397,607.82022 mm^4 |
Formula Used
J = πD⁴ / 32
J = π(Do⁴ − Di⁴) / 32
J = bh(b² + h²) / 12
J = πab(a² + b²) / 4
θ = TL / (GJ), only for circular torsion checks
Important note: for noncircular torsion, the torsion constant is generally more useful than the polar area moment.
How to Use This Calculator
- Select the cross-section shape.
- Choose a length unit for all geometric inputs.
- Enter the required dimensions for the selected shape.
- Add member length, torque, and shear modulus if needed.
- Click Calculate to show the result above the form.
- Review J in both chosen units and SI units.
- Inspect the Plotly graph for scaling sensitivity.
- Download the result summary as CSV or PDF.
Frequently Asked Questions
1. What does the polar moment of inertia measure?
It measures how area is distributed around a center point. Larger values usually mean better resistance to twisting for circular members.
2. Is this the same as mass moment of inertia?
No. This calculator reports the geometric polar moment of area. Mass moment of inertia depends on mass distribution and dynamics.
3. Why are circular sections common in torsion problems?
Circular sections distribute shear stress more uniformly in torsion. Their twist equations also use the polar moment directly.
4. Can I use the rectangle result for exact torsion design?
Not usually. Rectangular torsion is better handled with the torsion constant. The displayed J is still useful as a geometric property.
5. Why does the chart rise so quickly?
Polar moment scales with the fourth power of size during uniform scaling. Small diameter changes can produce very large stiffness changes.
6. Which units are best for reporting?
Use units consistent with your project. Mechanical reports often use mm^4 or in^4, while scientific work may prefer SI units.
7. When do I need shear modulus input?
Enter shear modulus only when you want advanced outputs such as torsional rigidity or angle of twist.
8. Does a hollow shaft always perform poorly?
No. Hollow shafts can be very efficient because more area sits farther from the center, which strongly improves torsional resistance.