# Angle of Twist Calculator

URL: https://speedycalculator.com/calculators/physics/angle-of-twist/
Last updated: 2026-10-30

> The angle of twist calculator finds how far a shaft twists under a torque, from its length, polar moment of inertia (solid or hollow round shaft) and shear modulus.

## Inputs

- Torque T (N·m, kN·m, lbf·ft, lbf·in, kgf·m)
- Shaft length L (m, cm, mm, km, in, ft, yd, mi)
- Cross-section
- Outer diameter D (mm, cm, m, in, µm)
- Inner diameter d (mm, cm, m, in, µm)
- Polar moment of inertia J (mm⁴, cm⁴, m⁴, in⁴)
- Material
- Shear modulus G (GPa, MPa, Pa, psi, Mpsi)

## Results

- Polar moment of inertia J
- Angle of twist φ
- In radians
- Maximum shear stress

## Formula

```
φ = T·L / (J·G),  J = π·d⁴/32 (solid) or π·(D⁴ − d⁴)/32 (hollow)
```

## Explanation

Drive shafts, torsion bars and screwdrivers all twist slightly when they carry torque. Too much twist causes vibration and poor control, so shafts are sized to keep it small – often below 1° per metre.

### How to use the Angle of Twist Calculator

- **Torque T**
- **Shaft length L**
- **Cross-section**
- **Outer diameter D**
- **Inner diameter d**
- **Polar moment of inertia J**
- **Material**
- **Shear modulus G**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Enter the torque and shaft length, choose the cross-section and the material (or a custom shear modulus).

### Inputs explained

FieldWhat to enterTorque TUnits: N·m, kN·m, lbf·ft, lbf·in, kgf·mShaft length LUnits: m, cm, mm, km, in, ft …Cross-sectionChoose one: Solid round shaft, Hollow round shaft, Custom polar moment JOuter diameter DUnits: mm, cm, m, in, µmInner diameter dUnits: mm, cm, m, in, µmPolar moment of inertia JUnits: mm⁴, cm⁴, m⁴, in⁴MaterialChoose one: Structural steel, Stainless steel, Aluminium alloy, Brass …Shear modulus GUnits: GPa, MPa, Pa, psi, Mpsi

### What each result means

- **Polar moment of inertia J** = [Polar moment of inertia]
- **Angle of twist φ** = deg([Torque T]×[Shaft length L]÷([Polar moment of inertia]×[G]))
- **In radians** = [Torque T]×[Shaft length L]÷([Polar moment of inertia]×[G])

### Example values

How the main result changes when **Torque T** changes and everything else stays at the example values above:

Torque T (N·m)Angle of twist φ (°)2500.86243751.29375001.72496252.15617502.58731,0003.4498

## Worked example

Inputs — Torque T: 500 N·m; Shaft length L: 1.2 m; Cross-section: Solid round shaft; Outer diameter D: 40 mm; Material: Structural steel.

Results — Polar moment of inertia J = 251,327.41229 mm⁴; **Angle of twist φ = 1.72489 °**; In radians = 0.030105 rad; Maximum shear stress = 39.78874 MPa.

## Key terms

- **Polar moment of inertia**: Resistance of a section to torsion.

## FAQ

**What limits shaft twist?**
Stiffness: GJ/L – larger diameters stiffen shafts dramatically (d⁴).

**Why are hollow shafts used?**
They save weight while keeping most of the stiffness.

**What is the shear modulus of steel?**
About 79–80 GPa.

**What is torsional rigidity?**
The product G·J.


## Sources

- [MIT OpenCourseWare – Mechanics of Materials](https://ocw.mit.edu/courses/2-001-mechanics-materials-i-fall-2006/)
- [The Engineering ToolBox](https://www.engineeringtoolbox.com/)