Polar Moment of Inertia Calculator

Enter a shaft's outer diameter (and inner diameter for a hollow tube) to find its polar moment of inertia (J), polar section modulus, and torsional shear stress under an applied torque.

Quick Facts

Solid shaft
J = πD⁴ / 32
D is the outer diameter; J grows with the 4th power of diameter.
Hollow shaft (tube)
J = π(D⁴ − d⁴) / 32
D = outer diameter, d = inner diameter.
Torsional shear stress
τmax = T·c / J
c = D/2 is the distance from the center to the outer surface.
Perpendicular axis theorem
J = Ix + Iy
For a planar cross-section, the polar moment equals the sum of the two rectangular (area) moments of inertia.

Your Results

Calculated
Polar Moment of Inertia (J)
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J = πD⁴/32 (solid) or π(D⁴−d⁴)/32 (hollow)
Polar Section Modulus (Zp)
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Zp = J / c, where c = D/2
Maximum Shear Stress (τmax)
-
τmax = T·c / J, from applied torque
Cross-Sectional Area (A)
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A = π(D² − d²)/4

Ready

Choose a shaft type, enter dimensions and torque, then press Calculate.

Formula and Method for Polar Moment of Inertia

The polar moment of inertia (J) measures a cross-section's resistance to torsion — twisting about its longitudinal axis. For a solid circular shaft, J = πD⁴ / 32, where D is the outer diameter. For a hollow circular shaft (a tube), J = π(D⁴ − d⁴) / 32, where D is the outer diameter and d is the inner diameter. Because diameter is raised to the 4th power, J grows very quickly as diameter increases — doubling the diameter increases J by a factor of 16.

How the calculation works

Choose whether the shaft is solid or hollow, then enter the outer diameter (and inner diameter for a hollow shaft) in a consistent unit. The calculator applies the formula above to get J. It also divides J by the outer radius c = D/2 to get the polar section modulus, Zp = J/c, which combines with an applied torque T to find the maximum shear stress at the outer surface: τmax = T·c/J = T/Zp. This shear stress is highest at the outer fiber of the shaft and zero at the central axis.

Common mistakes

  • Entering radius instead of diameter: this calculator (and most published J formulas) uses diameter, D. If you have a radius, double it first, or the result will be off by a factor of 16.
  • Confusing polar moment of inertia with area moment of inertia: J (polar) governs resistance to torsion (twisting); Ix and Iy (area/rectangular moments) govern resistance to bending. For a circular section they are related by the perpendicular axis theorem: J = Ix + Iy.
  • Forgetting the inner diameter on a hollow shaft: using the solid-shaft formula on a tube overstates J and understates the actual torsional stress.
  • Mixing units: keep the outer and inner diameter in the same unit before calculating; J's unit is that length unit raised to the 4th power (e.g., mm⁴ or in⁴).

Real-world applications

  • Drive shafts, axles, and propeller shafts are sized using J so torsional stress and twist stay within safe limits under the operating torque.
  • Torsion springs and torsion bars use J together with the shear modulus G to determine torsional stiffness (angle of twist per unit torque).
  • Structural and mechanical engineers use J to compare the torsional efficiency of solid versus hollow shafts — a hollow shaft can carry nearly as much torque as a solid one of the same weight, because material near the center contributes little to J.
  • Robotics and machine design use polar moment of inertia to size shafts and couplings that transmit rotational power without excessive twisting.

Frequently Asked Questions

What is the difference between polar moment of inertia and area moment of inertia?
Polar moment of inertia (J) measures resistance to torsion (twisting about the shaft's axis), while area (rectangular) moment of inertia (Ix, Iy) measures resistance to bending. For a planar cross-section they are related by the perpendicular axis theorem: J = Ix + Iy.
How do I find the polar moment of inertia of a hollow shaft?
Use J = π(D⁴ − d⁴) / 32, where D is the outer diameter and d is the inner diameter of the tube. Setting d = 0 reduces this to the solid-shaft formula, J = πD⁴ / 32.
How is polar moment of inertia used to find shear stress?
Maximum shear stress at the outer surface of a shaft under torque T is τmax = T·c / J, where c is the outer radius (D/2). A larger J for the same torque and radius produces a lower shear stress.
Does the polar moment of inertia depend on the shaft's material?
No. J is a purely geometric property of the cross-section's shape and size — it does not depend on what the shaft is made of. Material only enters later, through the shear modulus G, when calculating torsional stiffness or the angle of twist.

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