# Polar Moment of Inertia Calculator

URL: https://speedycalculator.com/calculators/physics/polar-moment/
Last updated: 2026-10-30

> The polar moment of inertia calculator finds J for a solid or hollow circular shaft, used to compute torsional stiffness, twist angle and shear stress, along with the polar section modulus.

## Inputs

- Shape
- Outer diameter D (mm, cm, m, in, µm)
- Inner diameter d (mm, cm, m, in, µm)

## Results

- Polar moment of inertia J
- Polar section modulus Zₚ
- Cross-section area

## Formula

```
Solid: J = πD⁴/32 · Hollow: J = π(D⁴ − d⁴)/32 · Z_p = 2J/D
```

## Explanation

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

FieldWhat to enterShapeChoose one: Solid circular section, Hollow circular sectionOuter diameter DUnits: mm, cm, m, in, µmInner diameter dUnits: 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)2538,349.519737.5194,144.443550613,592.315262.51,498,028.1132753,106,311.09551009,817,477.0425

## 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.

## FAQ

**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.


## Sources

- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)
- [The Engineering ToolBox](https://www.engineeringtoolbox.com/)