Torsion occurs when a body—such as a long solid cylinder of length $\ell$ and cross-sectional area $A$—is twisted by an applied torque. The relevant quantities and relationships are illustrated in the original diagram (not shown here).
As with shear, the twist angle $\gamma$ is proportional to the applied force, as long as the deformation remains elastic. Since torsion is essentially a form of shear, for small angles, the following relation also holds:
$$
\gamma = \frac{\tau}{G}
$$
However, in the case of torsion, the rotation angle $\varphi$ is of more interest. According to the geometry, it can be expressed as:
$$
\varphi = \frac{\Delta \ell}{r}
$$
where $\Delta \ell$ represents the linear displacement at the edge of the cylinder, and $r$ is the cylinder's radius.
From the same geometric sketch, for small $\gamma$, it follows:
$$
\gamma = \frac{\Delta \ell}{\ell}
$$
Substituting this into the expression for $\varphi$ and using the shear relation, we obtain:
$$
\varphi = \frac{\tau \ell}{G r}
$$
It is important to note that for a cylindrical object, the shear stress given by $F / (\pi r^2)$ must be multiplied by 4 to determine the torsional stress $\tau$.