Polar moment of inertia formula
The polar moment of inertia measures how well a round shaft resists twisting. For a solid circular shaft it is pi times the diameter to the fourth power divided by thirty-two, and it sets both torsional stiffness and shear stress.
Variables
| J | Polar moment of inertia | in⁴ or m⁴ |
| d | Shaft diameter (solid) | in or m |
Rearranged
Torsional shear stress: τ = T × r / J
Worked example
A solid shaft is 40 mm in diameter.
Apply J = π d⁴ / 32: π × 40⁴ / 32 = 251,327 mm⁴.
The polar moment governs torsion the way the area moment governs bending. Because diameter is raised to the fourth power, a small increase in shaft size sharply raises torsional stiffness. The shear stress in a twisted shaft is torque times radius divided by J.
Working a shaft in torsion?
See the Torque Formula and the Shear Stress Formula.
Polar moment and torsion
When a shaft carries torque, the polar moment of inertia decides how much it twists and how high the shear stress climbs. Stiffness rises with J, and the peak shear stress at the surface is torque times the outer radius divided by J. This is why drive shafts are sized on J, not just on cross-sectional area.
Solid vs hollow shafts
Because material near the center contributes little to J, a hollow shaft carries nearly the same torque as a solid one of the same outer diameter at a fraction of the weight. The polar moment of a hollow shaft is pi times the difference of the fourth powers of the outer and inner diameters over thirty-two, which is why power transmission shafts are often tubes.
FAQ
What is the polar moment of inertia formula?
For a solid round shaft, J equals pi times the diameter to the fourth power divided by thirty-two. For a hollow shaft, use the difference of the fourth powers.
What is the polar moment used for?
Torsion. It sets how much a shaft twists under torque and the shear stress it develops, found from T times r over J.
Why are hollow shafts efficient in torsion?
Material near the axis adds little to J, so removing it saves weight while keeping most of the torsional strength of a solid shaft of the same outer diameter.