I think that 'N-volume' including area, volume, and higher dimensional analogs is pretty understandable to a student. The fact that it's signed is more subtle,

I think that 'N-volume' including area, volume, and higher dimensional analogs is pretty understandable to a student. The fact that it's signed is more subtle, but still is not a big problem. The benefits to intuition of calling it 'volume' far outweigh the negatives. Anyway, 'scaling factor' still leaves the signs mysterious.