I'd never really thought of a story problem as being similar to a three act story. Maybe it's because I'm horrible at writing stories! But after reading examples on blogs such as Mr. Piccini's Mathtabulous Site and A Recursive Process, I think that volume problems in Geometry naturally lead to this type of set-up.
I taught Geometry again this past year, but this was the first year in a long time where I got to teach the second half of the material, which happens to be the half in our curriculum with Volume and Measurement. In fact, I think I've done a few "starter" or "warm-up" problems that could easily be rewritten into 3-act lessons.
For instance, I have done "which is larger?" problems in the past. Here's an example of a problem I wrote and gave to my class as a warm-up earlier this spring. This problem could easily be set up as a 3-act story problem. I'm certain the students would be much more engaged if I gave the set-up first, added the information and back-story, then finally gave the "reveal" at the end. I was intrigued by the idea of adding a "sequel" as well, or letting the students come up with their own math problems.
If I'm teaching Geometry this year, I'll definitely add some problems like these to my curriculum!