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I want to know the depth of a guitar neck (distance from top of fretboard to back of neck). I don't have any calipers, and I don't need calipers aside from this very specific purpose so I super do not want to buy some. So I wrapped a piece of paper around the back of the neck and marked the fretboard edges on it to get the measurement from one side to the other. Would this number (2.56"/6.6cm in this case) be useful at all in determining the neck depth? I got a C- in high school geometry, to explain my total inability to figure this out on my own.

Won't really work. Reason(s) why is:

1.Theres a good chance that your neck profile isn't an arc of a perfect circle, so you can't use pi as a reference in determining the measurement.
2.Even if your neck profile was a section of a perfect circle, you would need to know the radius of the arc section at the spot that you measured from.

How dead on does the measurement need to be...? If you have a carpenters square, you can set the neck on its side and put the square against the front and back side of the neck and then measure that distance. Or even simpler, take two pencils or chop sticks--any two straight, stick or rod type same things. Tape one perpendicular to the top (fretboard) and one perpendicular to the back. Make sure they are parallel and measure the distance between them.

The stick method is brilliant! I figured out it's roughly 21mm. Close enough for my purposes. Thank you!

No problem....Glad it worked for you..!!