It only wouldn't be a cheat if the question were framed as "Surprisingly, the volume of the ring is constant, regardless of the radii of the sphere and circle.

It only wouldn't be a cheat if the question were framed as "Surprisingly, the volume of the ring is constant, regardless of the radii of the sphere and circle. What is that constant volume?" Otherwise, you don't really know your answer is correct. Maybe, they wanted the answer in terms of R1 and R2. And if you are able to trick them into confirming it, then you're essentially using social engineering to leverage somebody else's work (the person who discovered the surprising result) in formulating your answer. There's no way that the original discoverer could have used that method. That's what makes it a cheat. Back in middle school geometry, I came up with my own strategy for proving theorems. "You wouldn't ask me to prove it if it weren't true, therefore it must be true. Q.E.D." It didn't go over well.