The calculation itself would be easy to do IF you could reference the volume. Unfortunately, I don't think you can use the volume inside a user parameter without some sort of VBA programming. The volume is a property, which is a different animal kept separate from the parameters.
The only thing I could think of besides VBA would be to brute-force one big formula as the sum of each triangular and rectangular area. If everything was modeled in Inventor, you should theoretically have every parameter necessary to calculate the area.
If one formula gets too large, you can make multiple intermediate user parameters, then add them together in yet another parameter.
Unfortunately, as soon as you change the shapes of any of the pieces, the formula would need to be revised to stay accurate. If it's always going to consist of the same shapes, it wouldn't take a terrible amount of time to go through with this.
If there are more elegant ways to accomplish the same thing, by all means do those first.