By Zencalcy Editorial TeamLast reviewed About us
The polar moment of inertia measures a circular section's resistance to twisting. For a solid shaft it is πd⁴/32; for a hollow shaft it is π(D⁴ − d⁴)/32. It appears directly in the torsion formula for shear stress and angle of twist.
Torsional resistance of solid and hollow circular sections.
Solid: J = πd⁴ / 32 Hollow: J = π(D⁴ − d⁴) / 32
Both diameters must use the same unit. The hollow formula reduces to the solid one when the inner diameter is zero.
| Symbol | Meaning |
|---|---|
| D | Outer diameter, in metres. |
| d | Inner diameter, in metres. |
| J | Polar moment of inertia, in metres to the fourth. |
A 50 mm solid shaft has J = π × 0.05⁴ / 32 = 6.136×10⁻⁷ m⁴, or about 613,600 mm⁴.
Continue with the Projectile Motion Calculator, Projectile Range Calculator, or the Quarter Mile Calculator — or browse more kinematics tools and all physics calculators.
The polar moment is about the shaft axis and resists torsion. The area moment about a diameter resists bending.
Results are computed from the standard equations of classical mechanics. Unit conversions use the SI definitions published by NIST. Air resistance is ignored unless the calculator names a drag model.
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Content updated: September 2026