The polar moment of inertia (polar second moment of area) measures a cross-section’s resistance to twisting. Doubling a shaft’s diameter makes it 16 times stiffer in torsion, which is why drive shafts are made as wide tubes.
How to use the Polar Moment of Inertia Calculator
- Shape
- Outer diameter D
- Inner diameter d
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose solid or hollow and enter the outer (and inner) diameter. J is used in τ = T·r/J and φ = T·L/(G·J).
Inputs explained
| Field | What to enter |
|---|---|
| Shape | Choose one: Solid circular section, Hollow circular section |
| Outer diameter D | Units: mm, cm, m, in, µm |
| Inner diameter d | Units: mm, cm, m, in, µm |
Example values
How the main result changes when Outer diameter D changes and everything else stays at the example values above:
| Outer diameter D (mm) | Polar moment of inertia J (mm⁴) |
|---|---|
| 25 | 38,349.5197 |
| 37.5 | 194,144.4435 |
| 50 | 613,592.3152 |
| 62.5 | 1,498,028.1132 |
| 75 | 3,106,311.0955 |
| 100 | 9,817,477.0425 |
Formula
Solid: J = πD⁴/32 · Hollow: J = π(D⁴ − d⁴)/32 · Z_p = 2J/D
Worked example
Inputs — Shape: Solid circular section; Outer diameter D: 50 mm.
Results — Polar moment of inertia J = 613,592.3152 mm⁴; Polar section modulus Zₚ = 24,543.6926 mm³; Cross-section area = 1,963.4954 mm².
Key terms
- Polar moment of inertia
- ∫r² dA over the cross-section.
Frequently asked questions
Why are hollow shafts used?
Material near the centre adds little torsional stiffness, so removing it saves weight.
What is the difference from the area moment of inertia?
J = Iₓ + I_y, measured about the axis of the shaft.
How does J affect twist?
The twist angle is inversely proportional to J.
What units does J have?
Length to the fourth power, e.g. mm⁴.