Overlapping Solids

The orientation of the torus is irrelevant. We are told that the volumes of $A \cap B'$ and $A' \cap B$ are equal. Adding the volume of $A \cap B$ to each of these, this means that the volumes of $A$ and $B$ are equal. Since the volume of the enneadecagonal prism $A$ is $4 \times 19 \times \tfrac12 \times s \times \frac{s}{2\tan\frac{\pi}{19}} \; = \; 19s^2 \cot\tfrac{\pi}{19}$ and the volume of the torus $B$ is $2\pi^2 \times 2 \times \big(\tfrac{4}{\pi}\big)^2 \; = \; 64$ we deduce that $s \; = \; \sqrt{\tfrac{64}{19}\tan\tfrac{\pi}{19}} \; = \; \boxed{0.7497266561}$