How does a Bourdon tube maintain constant volume?

I would like to understand why a Bourdon tube works, the working principle is described on wikipedia as:

The Bourdon pressure gauge uses the principle that a flattened tube tends to straighten or regain its circular form in cross-section when pressurized. This change in cross-section may be hardly noticeable, involving moderate stresses within the elastic range of easily workable materials. The strain of the material of the tube is magnified by forming the tube into a C shape or even a helix, such that the entire tube tends to straighten out or uncoil elastically as it is pressurized.

I have searched far and wide for a detailed explanation and have not found one. Is it possible to show (without too too much hassle) that the volume change in the tube is negligible, and maybe an approximate formula for the amount of straightening the tube will experience for a given pressure?