The angle of twist of a circular shaft caused by torsional deformation is a function of the internal torque, the length of the shaft, its polar moment of inertia and the shear modulus of the material:
This formula is valid only if the internal torque and the cross-sectional area are constant over the length. Also, the material has to be homogeneous. It is not valid for thin walled tubes. The angle of twist is measured in radians.
If one or more of these is not constant, it should be written as a function of the location on the shaft. The angle of twist equation can then be obtained by solving the integral: