What is work done by torque?

In my textbook it's written Work done is always calculated on the displacement of the body parallel or anti-parallel to the direction of the force. And, according to the formula it is : $\mathbf{F \cdot S}$ or $Fs \cos \theta$ And work done by torque is given as : $\tau \theta$ Now let's take a case when a cylinder with fixed center of mass and radius $R$ is rotated with a tangential force $F$ And it rotates $\pi$ radians. So, according to the first formula, the displacement is $2R$ but since the the point of application of force has moved , it is right vertically downwards from the initial point of application. So, the angle with the point of application of force and the force vector becomes $\frac{\pi}{2}$ And $\cos \frac{\pi}{2}= 0$ . And so $W = 0$ . But the second formula says otherwise, it says that work done is $\tau \theta$ So $W= \pi FR$ according to second formula. So yeah, I need to know which is right.