Angle of Twist Calculator

Find the twist angle of a circular shaft under torque using θ = T·L / (J·G), along with the polar moment of inertia and the maximum shear stress at the outer surface.

Quick Facts

Formula
θ = T·L / (J·G)
T=torque, L=length, J=polar moment of inertia, G=shear modulus
Solid shaft J
J = π·d⁴ / 32
Use outer/inner diameters for a hollow shaft instead
Assumption
Linear-elastic, homogeneous material
Valid only while shear stress stays below the proportional limit

Your Results

Calculated
Angle of twist (θ)
-
Total twist over the shaft length, in degrees
Angle of twist (radians)
-
SI unit form used in θ = TL/JG
Polar moment of inertia (J)
-
Cross-section resistance to torsion
Max shear stress (τ)
-
At the outer surface, τ = T·r / J

Ready

Enter torque, shaft length, diameter, and material, then press Calculate.

Understanding Angle of Twist

When a torque is applied to a shaft, the shaft rotates by an angle called the angle of twist. For a straight, circular shaft made of a linear-elastic, homogeneous material, this angle can be found directly from the shaft's geometry and material properties, without any physical testing.

The formula

The standard torsion formula relates the angle of twist θ (in radians) to the applied torque, the shaft's geometry, and its material:

θ = T · L / (J · G)

  • T — applied torque, in newton-meters (N·m)
  • L — length of shaft over which the torque acts, in meters (m)
  • J — polar moment of inertia of the cross-section, in m⁴; for a solid circular shaft of diameter d, J = π · d⁴ / 32
  • G — shear modulus (modulus of rigidity) of the shaft material, in pascals (Pa)

The result comes out in radians; multiply by 180/π to convert to degrees, which is usually easier to picture. This calculator also reports the maximum shear stress at the shaft's outer surface, τ = T · r / J, where r = d/2 is the outer radius — a useful check that the shaft is staying within its elastic range.

Working with units

  • This calculator takes torque in N·m, length in m, diameter in mm (converted internally to meters), and shear modulus in GPa (converted to pascals) via the material selector.
  • Typical shear modulus values: steel ≈ 79 GPa, aluminum alloys ≈ 26 GPa, brass ≈ 40 GPa, titanium alloys ≈ 44 GPa, copper ≈ 48 GPa.
  • For a hollow shaft, replace J with π · (do⁴ − di⁴) / 32 using the outer and inner diameters; this calculator assumes a solid circular cross-section.

Knowing the limits

This formula assumes the shaft is straight, has a uniform circular cross-section along its length, and stays within the linear-elastic range of the material — meaning the shear stress never exceeds the proportional limit. It does not apply to non-circular cross-sections (which warp out of plane under torsion and need a shape-specific formula), tapered shafts, or shafts loaded beyond yield.

Frequently Asked Questions

What is the formula for angle of twist?
The angle of twist for a circular shaft is θ = T·L / (J·G), where T is the applied torque, L is the shaft length, J is the polar moment of inertia of the cross-section, and G is the shear modulus of the material. θ comes out in radians; multiply by 180/π to convert to degrees.
How do I find the polar moment of inertia J?
For a solid circular shaft, J = π·d⁴ / 32, where d is the shaft diameter. For a hollow shaft, use J = π·(do⁴ − di⁴) / 32 with the outer diameter do and inner diameter di.
What shear modulus should I use?
Shear modulus (G) depends on the material: roughly 79 GPa for steel, 26 GPa for aluminum alloys, 40 GPa for brass, 44 GPa for titanium alloys, and 48 GPa for copper. Use a value specific to your actual alloy when precision matters.
Does this formula work for non-circular shafts?
No. θ = TL / (JG) assumes a solid or hollow circular cross-section, the only common shape that does not warp out of its plane under torsion. Square, rectangular, and other open sections twist according to different, shape-specific formulas.

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