What this calculator does
Polar moment of inertia measures how a cross-section resists twisting, the rotational counterpart to the way an ordinary second moment of area measures resistance to bending. It is a purely geometric property of the shape, in units of length to the fourth power, and it does not depend on the material or the load applied to it.
This calculator works out the polar moment of inertia for the two shapes that come up most often in shaft and torsion design, a solid circular cross-section and a hollow circular tube, along with the polar section modulus used to convert an applied torque directly into shear stress.
The formula
For a solid circular shaft, polar moment of inertia is J = πd⁴/32, where d is the outer diameter. For a hollow tube, the inner circle’s contribution is subtracted: J = π(dₒ⁴ − dᵢ⁴)/32. The polar section modulus, Zₚ = J/(dₒ/2), then converts an applied torque T into maximum shear stress at the outer surface via τ = T/Zₚ.
| Term | Meaning |
|---|---|
| J | Polar moment of inertia, a geometric measure of resistance to twisting, in length⁴. |
| dₒ, dᵢ | Outer diameter and, for a hollow section, inner diameter. |
| Zₚ | Polar section modulus, J divided by the outer radius, used to find shear stress from an applied torque. |
The inputs explained
| Field | What to enter |
|---|---|
| Cross-section | Choose a solid circular cross-section, or a hollow circular tube. |
| Outer diameter (mm) | The outer diameter of the shaft or tube. |
| Inner diameter (hollow only) (mm) | The inner diameter, for a hollow tube only. Must be smaller than the outer diameter. |
When to use it
Sizing a shaft against torsion
A drive shaft, axle or torsion bar transmitting rotational power needs its polar moment of inertia to relate an applied torque to the resulting shear stress and angle of twist, the core check in torsion design.
Comparing a solid shaft against a hollow tube
A hollow tube removes material from the centre, where it contributes least to twisting resistance, so a tube can match a solid shaft’s polar moment of inertia at a noticeably lower weight, a common trade-off in aerospace and bicycle-frame design.
Checking a moment of inertia formula by hand
A mechanics of materials problem giving a shaft diameter, or an outer and inner diameter for a tube, is solved directly here, without separately raising each diameter to the fourth power and dividing by 32 by hand.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Polar moment of inertia across a range of solid shaft diameters
A solid circular shaft, across a range of outer diameters.
| Diameter | Polar moment of inertia J | Polar section modulus Zₚ |
|---|---|---|
| 20 mm | 15,707.96 mm⁴ | 1,570.7963 mm³ |
| 30 mm | 79,521.56 mm⁴ | 5,301.4376 mm³ |
| 40 mm | 251,327.41 mm⁴ | 12,566.37 mm³ |
| 50 mm | 613,592.32 mm⁴ | 24,543.69 mm³ |
| 60 mm | 1,272,345.02 mm⁴ | 42,411.50 mm³ |
How much does hollowing out a shaft reduce its polar moment of inertia?
A fixed outer diameter of 50 mm, across a range of inner diameters for a hollow tube.
| Inner diameter | Polar moment of inertia J | Polar section modulus Zₚ |
|---|---|---|
| 10 mm | 612,610.57 mm⁴ | 24,504.42 mm³ |
| 20 mm | 597,884.35 mm⁴ | 23,915.37 mm³ |
| 30 mm | 534,070.75 mm⁴ | 21,362.83 mm³ |
| 40 mm | 362,264.90 mm⁴ | 14,490.60 mm³ |
| 45 mm | 211,014.40 mm⁴ | 8,440.5759 mm³ |
Questions
What is the formula for polar moment of inertia?
For a solid circular cross-section, J = πd⁴/32. For a hollow circular tube, J = π(dₒ⁴ − dᵢ⁴)/32, subtracting the inner circle’s contribution from the outer one.
What are the units of moment of inertia here?
Polar moment of inertia is a purely geometric quantity with units of length to the fourth power, shown here in mm⁴ for typical shaft dimensions entered in millimetres. It carries no mass or force units, unlike a rotational (mass) moment of inertia.
How is polar moment of inertia different from the mass moment of inertia used in rotational dynamics?
Polar moment of inertia describes a cross-section’s resistance to twisting under torque, in length⁴, independent of mass. Mass moment of inertia describes a rotating body’s resistance to angular acceleration, in mass×length², and depends on how mass is distributed. They answer different engineering questions despite the similar name.
Why does a hollow tube resist twisting almost as well as a solid shaft of the same outer diameter?
Material near the centre of a circular cross-section contributes little to polar moment of inertia, since the formula scales with the radius to the fourth power. Removing that central material to make a tube costs relatively little twisting resistance for a large saving in weight.
For the mass moment of inertia of a rotating solid shape, used in rotational dynamics rather than torsion, see the moment of inertia calculator. For a circle’s basic area and circumference, see the circle calculator.