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
| Field | What to enter |
|---|---|
| Torque T | Units: N·m, kN·m, lbf·ft, lbf·in, kgf·m |
| Shaft length L | Units: m, cm, mm, km, in, ft … |
| Cross-section | Choose one: Solid round shaft, Hollow round shaft, Custom polar moment J |
| Outer diameter D | Units: mm, cm, m, in, µm |
| Inner diameter d | Units: mm, cm, m, in, µm |
| Polar moment of inertia J | Units: mm⁴, cm⁴, m⁴, in⁴ |
| Material | Choose one: Structural steel, Stainless steel, Aluminium alloy, Brass … |
| Shear modulus G | Units: 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 (°) |
|---|---|
| 250 | 0.8624 |
| 375 | 1.2937 |
| 500 | 1.7249 |
| 625 | 2.1561 |
| 750 | 2.5873 |
| 1,000 | 3.4498 |
Formula
φ = T·L / (J·G), J = π·d⁴/32 (solid) or π·(D⁴ − d⁴)/32 (hollow)
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.
Frequently asked questions
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.