I haven’t done much measuring of bevel angles, but hope to do so in the future.
Here is a brief list of ways I’ve heard about for measuring bevel angles:
(1) Sharpie Method: Use a Sharpie marker on the entire bevel, then adjust your sharpener so that it removes all of the marker ink. Then measure the angle of your sharpener.
(2) Laser Goniometer: Buy a laser goniometer from CATRA, or make one yourself. This bounces a laser off the knife edge, and measures the bevel angle from the angle of the reflected beam. You can also make one yourself using a laser-pointer.
CATRA sells several models, here is one:
http://www.catra.org/pages/products/kniveslevel1/lgpm.htm
Or you can make your own (see Appendix):
http://www-archive.mse.iastate.edu/fileadmin/www.mse.iastate.edu/static/files/verhoeven/KnifeShExps.pdf
(3) Make a Mold: Spray the knife blade with a mold-release agent, or WD-40. Then make a mold of the knife edge, say using epoxy-putty or anything that will be rigid enough. After the mold sets, remove the knife, and then cut/saw a cross section of the mold. Smooth/polish the cross section, and take a photo of it under a microscope. Measure the angle in the image.
There are probably many other ways. I suspect that making a mold is the most reliable way, but probably the most work. To be super-precise, you will need the plane of the cross-section to be perpendicular to the edge of the knife. The laser goniometer sounds neat, but the reflected laser beam might need some interpreting if the beam spreads out due to micro-scratches or convexing.
Sincerely,
–Lagrangian