Calculating the Bevel Angle

[bevansmw]

Just finished taking Calc I and was thinking about some trig principles today while showering and getting ready to shave. I thougth about honing and realized I could figure out the actual bevel angle put on a razor using trig. It's a pretty simple calculation if you're familiar with trig, here is how it works.

A straight razor laying flat against a hone establishes a right triangle, with a points of contact with the hone at the edge and the spine. By taking the spine width and dividing it by two, you get an approximate length of the opposite side of the triangle (side standing up), while the blade width provides the length of the hypotenuse (long side of triangle). By using the sin theta you don't need to calculate the bottom of the triangle as sin theta = opposite / hypotenuse. This allows us to get the actual angle between the hypotenuse and the adjacent side or bottom of the triangle (hone would be the adjacent side). I divided the spine width by two in order to get the angle from the exact center of the blade, where the very tip of the edge will be after honing. The angle would be exactly what would be established for the bevel after honing. By taking the arcsin of the fraction you get the angle because the formula reads...

sin theta = (opposite / hypotenuse)

Therefore...

arcsin(opposite / hypotenuse) = theta

theta = angle

Running some basic numbers you get the following bevel angles...

Spine Width......Blade Length.....Bevel Angle
3/16"..............5/8"................8.63 degrees
1/4"...............5/8"................11.54 degrees
3/16"..............3/4"................7.2 degrees
1/4"................3/4"...............9.6 degrees
3/16"...............7/8"..............6.2 degrees
1/4"...............7/8"................8.2 degrees
3/16"...............1".................5.4 degrees
1/4"...............1"..................7.2 degrees

Some generalizations can be made from the information gathered here...

As the spine gets thicker the bevel angle increases.

As the blade gets longer the bevel angle decreases.

A smiling edge will have a bevel angle that varies, being largest at the shortest part of the blade and smallest at the longest part of the blade. Between these points the bevel angle will vary from a greater angle to smaller angle and back to a greater angle, analyzing the blade from toe to heel or vice versa.

As an example, a smiling blade with 5/8" toe, 3/4" center and 5/8" heel with 3/16" spine will have a bevel angle varying from approximately 8.63 degrees at the toe to 7.2 degrees at the center and back to 8.63 degrees at the heel.

Wedges work a bit different if the entire blade rests against the hone while honing. If that is the case then the triangle's opposite side (standing up side) would go the entire length of the spine. If the blade does not rest against the hone and points of contact are only the spine and edge then the previous calculations using 1/2 the spine width work.

Just thought I'd throw this out there