A shaft has two ways to be wrong. It can break — that is the stress check you just did — or it can simply wind up too far, so that the thing on the far end arrives late and out of position. The second failure has its own relation: $\varphi = \dfrac{T L}{J G}$, read aloud phi equals T L over J G, with $\varphi$ the Greek letter phi.

Name every letter before it works. $\varphi$ is the angle of twist between the two ends, and it comes out in radians. $T$ is the applied torque and $L$ is the length of shaft being twisted — the free span between whatever holds one end and whatever loads the other. $J$ is the polar moment of area, and $G$ is the shear modulus of the material, its stiffness in shear, which for structural steel is 80 GPa = 80 000 N/mm². What twists the shaft sits on top; what resists the twisting sits underneath. That is the whole shape of it.

Radians are not a decorative detail. Work the arithmetic in N·mm, mm, mm⁴ and N/mm² and every unit cancels — the answer is a bare number, and a bare number of that kind is an angle in radians, because a radian is defined as arc divided by radius. Nothing in the algebra has ever heard of a degree. So the last line of the working is always $\times \dfrac{180}{\pi}$, roughly 57.3, and the number should get BIGGER. If it got smaller you multiplied by π/180 and are now 3283 times out.

The trade rule worth carrying: about one degree of wind-up per 20 diameters of length. It is a rule of thumb, not a code, but it will tell you in three seconds whether a shaft that passed on strength is going to embarrass you on stiffness.