Polar Moment of Inertia Calculator
Calculate the polar moment of inertia of a solid circular shaft from its diameter, used for torsional stiffness and shear stress calculations.
How this calculator works
When a shaft transmits torque — turning a drive shaft, an axle, or a motor coupling — it twists rather than bends. Polar moment of inertia (J) describes how much a round cross-section resists that twisting, based only on its diameter.
This calculator takes the diameter of a solid circular shaft and returns J, which then feeds into torsional shear stress and angle-of-twist calculations.
Formula: For a solid circular shaft, polar moment of inertia J = π × diameter⁴ ÷ 32.
Worked example
A 2-inch diameter shaft:
- J = π × 2⁴ ÷ 32 ≈ 1.571 in⁴
Notes
J measures a round shaft’s resistance to twisting (torsion), the rotational counterpart to the bending moment of inertia I. It’s used to find torsional shear stress and angle of twist in rotating shafts, axles, and torque-transmitting members.
Diameter matters enormously here, since it’s raised to the 4th power in the formula. A shaft twice as thick has 16 times the polar moment of inertia — a small increase in diameter buys a large increase in torsional stiffness.
How to use
Enter the diameter of your solid circular shaft, in inches. The calculator returns the polar moment of inertia (J), which you can carry into the Shear Stress Calculator to find torsional shear stress under a given applied torque.
This is a planning estimate using standard mechanics formulas, not a substitute for engineering design. Confirm actual member sizing and code compliance with a licensed structural engineer.
Frequently asked questions
What is polar moment of inertia used for?
Polar moment of inertia (J) measures a round shaft's resistance to twisting, or torsion. It's the rotational counterpart to the bending moment of inertia (I), and it's used to calculate torsional shear stress and angle of twist in rotating shafts, axles, and other torque-transmitting members.
How is polar moment of inertia different from regular moment of inertia?
Regular moment of inertia (I) describes resistance to bending. Polar moment of inertia (J) describes resistance to twisting around the shaft's central axis. Both depend only on cross-section geometry, but they're used in different formulas — I with bending stress and deflection, J with torsional shear stress and twist angle.
Why does shaft diameter matter so much for torsional stiffness?
Because diameter is raised to the 4th power in the polar moment of inertia formula, small increases in diameter produce large increases in torsional stiffness. A shaft twice as thick has 16 times the polar moment of inertia, which is why designers reach for a modest diameter increase before other fixes when a shaft twists too much.
Does this formula work for hollow shafts?
No — this calculator's formula (J = π × diameter⁴ ÷ 32) is for a solid circular shaft only. A hollow shaft (a tube) uses a different formula that subtracts the inner diameter's contribution from the outer diameter's, since the missing material at the center no longer contributes to torsional resistance.
Estimates only. Verify quantities with your supplier before purchasing.