The Bevel Angle, Exposed and Revealed

[Slash McCoy]

There seems to be some confusion as to exactly how the bevel
angle of a razor is measured. I will call everyone's
attention to the basic formulae for solving a right triangle.
(A triangle having one 90 degree angle)

Sine of the angle = Opposite side / Hypotenuse
Cosine of the angle = Adjacent side / Hypotenuse
Tangent of the angle = Opposite side / Adjacent side

These formulae only hold for a right triangle. They may not
be applied to an isosceles triangle. To solve an isosceles
triangle, it must be broken into two right triangles, which
can be solved using the simple formulae above. To measure and
calculate using just the spine thickness and the blade width
is wrong on several counts. First of all, it assumes a right
triangle, and the spine thickness and blade width actually
form an isosceles triangle. Second, the thickest part of the
spine is not separated from the edge by the width of the
blade, but by a distance somewhat less than that. Third, the
thickest part of the spine might not in fact lie on the bevel
plane, but could conceivably lie somewhere inside the bevel.
And I have been as guilty as everyone else in using imprecise
methods of calculation using irrelevant measurements.

.

.

The illustration is a rough and ready representation of a
full hollowground razor. Notice the flat bevel at the edge,
and a flat bevel on the spine which is obviously the same
bevel. More or less to scale, the razor is a 6/8, with a
spine thickness of about 11/64". The bevel angle is 15
degrees. Ohhh, sharp, huh? Now, we want to calculate the
bevel angle, right? We must split the isosceles triangle into
two right triangles along the center plane of the blade. Each
triangle will then have 1/2 the spine thickness as the
OPPOSITE SIDE. Now we need either the ADJACENT or the
HYPOTENUSE to solve for either Sine or Tangent. Obviously,
measuring the ADJACENT side accurately will be awkward. I
prefer to measure the HYPOTENUSE. The best way I have found
to do this is to select a point on the spine and the edge an
inch or so from the tip. Hit it with a sharpie, trying to
leave the ink wet. Quickly before the ink dries, rest the
blade tip on a piece of flat stretched paper so that the
spine and edge are both in contact. Rock the blade down so
the sharpie ink on the spine and the edge make an imprint on
the paper. Then it is a simple matter to measure the greatest
distance between the marks with a dial caliper.

The overall blade width in the illustration is 600 pixels.
Scaled, that is 6/8 or 3/4". But remember, that measurement
is irrelevant. The ADJACENT side of the triangle is 532
pixels long in the illustration, but in the real world it
would be awkward to measure precisely. The hypotenuse is 539
pixels long in the illustration, about 43/64". The OPPOSITE
side happens to be half the spine thickness in this example,
11/128". Going with the Sine formula, S = 0 / H, so the Sine
= 11/128 / 43/64 = 0.1279, and the inverse sin function on
the calculator says that this is the sine for an angle of
7.34 degrees. Doubling this to get the entire bevel angle
gives a calculated bevel of 14.7 degrees. A little off, but
write it off to the pixels and the twidgety widgety up and
down scaling and stuff. .3 degrees is actually kind of small
potatoes, anyhow. A real world measurement, accurately taken,
then properly calculated, will probably give a much more
accurate determination of bevel angle. Oh, and my drawn bevel
line could have been a bit off, too. Darn pixels.

Now, if you had done what many do, and divided the spine
thickness by the blade width, you would have got a sine of
0.229 and a bevel angle of 13.2 degrees. Quite a difference,
huh? One angle should shave. The other would probably not
keep an edge worth a darn. And horrors... you might think the
spine should get about 4 layers of tape when honing! Of
course this would give you a secondary compound bevel of
about 18 degrees... still sort of shave-able but the actual
15 degree original bevel was pretty close to ideal.

Please, anybody with anything to add go ahead and throw your
two centavos in the ring.

 

[Legion]

I had a flashback to year eight maths class...

And when I woke up my beer was warm.

 

[tomjr]

beats my 2 pair.

 

[kiwichris]

Sounds like too much hard work to me. The big question is ... is it sharp enough to remove whiskers?

 

[Brownbear]

For the trig challenged. The widths are scaled to the normal blade widths, but note that this is not the same as the distance from the edge to the place where the spine contacts the hone, which is what determines the angle. I also didn't compensate for the fact that the width measured at the blade face is a trifle longer than the width measure at the center of the thickness, shouldn't matter much. If you find errors let me know.

 

[cpcohen1945]

Easier might be:

. . . Measure the width of the blade from the thickest part of the spine to the edge (call that "Width");

. . . Measure the thickness of the spine at its thickest point with a micrometer (call that "Thickness");

Use this formula:

. . . Bevel angle = 2 x arcsine ( (Thickness / 2) / Width ) )

The "thickest part of the spine" is the part of the spine, farthest from the edge, that contacts the hone during honing.

. Charles

 

[mycarver]

I'm gonna need somebody's notes if there is a pop quiz at the end of this. I wandered off for my roll of electrical tape and honed up two more blades during all this.

 

[Pilotgod]

I was told there would be no math.

J/K Thanks for explanation. I have been comtemplating the move to straights for a while and reading the wiki and numerous threads, but honing, bevels, and grinds are definitely the most challenging parts for me to wrap my head around.

 

[Seraphim]

I was told there would be no math.

J/K Thanks for explanation. I have been comtemplating the move to straights for a while and reading the wiki and numerous threads, but honing, bevels, and grinds are definitely the most challenging parts for me to wrap my head around.

Once you get a razor in hand, it all starts to make sense. Much of the chatter is minutiae that is of little practical effect.

 

[mycarver]

Once you get a razor in hand, it all starts to make sense. Much of the chatter is minutiae that is of little practical effect.

Phew,,that's a relief. I was just about to start measuring all my razors and get rid of the junkers that don't "measure up" . But I have to say , I now have one more weapon to use at the Antique Mall to get a lower price on a razor by making a few quick calculations. Then again if the angle isn't there,, I just don't want that Gold Edge since the angle is off a half a degree anyway. Lucky me that I didn't blow that $20.00.

 

[Seraphim]

 

[ouch]

These formulae only hold for a right triangle. They may not
be applied to an isosceles triangle. .

I see you never watched The Wizard of Oz.

 

[rajagra]

Please, anybody with anything to add go ahead and throw your
two centavos in the ring.

You refer to illustrations that aren't shown in your post.

 

[Seraphim]

 

[mycarver]

Now I'm not into all the mental mast,,, er,,lets say,,, gymnastics,, but from my experience once the wear on the spine exceeds the width of the edge your honing the edge will wear faster if even pressure is applied. Seems logical that a section, say, 1/16 will offer more resistance to wear than the hairline edge you're making. So all the theories go out the window as even pressure won't create even wear over the "life" of the blade. So the idea angle will only occur for a moment in time. Once that moment is reached the edge will wear away faster and wreak havoc with the cherished angles. Now you have to just throw the razor away.

And since most of my blades run between 11 and 13/64ths.and have different blade thicknesses,, and widths not dependent on any other factors, by default,, many are simply junk according to the prevailing theories since all angles will be different to some degree. I'm simply in shock that I can actually shave with any of them.
Well in my line of work I'm dealing with sharp edges regularly and at different sites they'll argue ad nauseum with great conviction about the ideal angles for carving. Me,, I just sharpen 'em up and go about using my tools and getting something done. Now,, I'm gonna hone up some razors with what I'm sure aren't the ideal angles and enjoy the shave anyway.

One other thought,, I don't think rocket scientists were designing these things 100 yrs. ago. Were angles their greatest concern? Naww, punch 'em out , hone 'em up put pretty scales on 'em and off they go. And here we are ,, years later buying them up like fools.

 

[mycarver]

And ,, a bit of a historical foot note. Granted, way back when ,electrical tape wasn't needed for the razors as they were fairly new. But after 50-75 yrs of hone wear, in 1946 (?) 3M marketed Spine Tape to help these ailing razors to equalize the wear and let the razor glide over the stones. Electricians just happened upon it and found it worked great in their trade. Over time , straights fell out of favor because of some nut named King Gillette, and the name gradually changed to "Electrical Tape" to cater to a much larger market.

 

[Slash McCoy]

Now I'm not into all the mental mast,,, er,,lets say,,, gymnastics,, but from my experience once the wear on the spine exceeds the width of the edge your honing the edge will wear faster if even pressure is applied. Seems logical that a section, say, 1/16 will offer more resistance to wear than the hairline edge you're making. So all the theories go out the window as even pressure won't create even wear over the "life" of the blade. So the idea angle will only occur for a moment in time. Once that moment is reached the edge will wear away faster and wreak havoc with the cherished angles. Now you have to just throw the razor away.

And since most of my blades run between 11 and 13/64ths.and have different blade thicknesses,, and widths not dependent on any other factors, by default,, many are simply junk according to the prevailing theories since all angles will be different to some degree. I'm simply in shock that I can actually shave with any of them.
Well in my line of work I'm dealing with sharp edges regularly and at different sites they'll argue ad nauseum with great conviction about the ideal angles for carving. Me,, I just sharpen 'em up and go about using my tools and getting something done. Now,, I'm gonna hone up some razors with what I'm sure aren't the ideal angles and enjoy the shave anyway.

The proof is in the pudding, as they say. If it shaves, it shaves. If it doesn't even after honing, or the edge seems too harsh or fragile, then examination of the bevel angle may offer clues as to why. Also, consideration of the bevel angle is a valid determining factor in making an INFORMED and REASONED decision to tape the spine or not, when honing.

And yes, you bring up a valid point on spine wear slowing down as the spine bevel increases in surface area. However, this is partially offset by the somewhat softer steel of the spine. Over the life of the razor, the speed at which the spine wears vs the speed at which the edge wears, probably averages out to give tolerably consistent results. I have measured very old and severely used and honed razors that were still holding a bevel of a reasonable angle. The straight razor as we know it evolved in the hands of observant and meticulous craftsmen to take best advantage of the steel and its properties. When you just HONE it, it usually will work. It is designed to do so. Mostly, it is only when the razor is used or maintained outside the original expected envelope of methodology that disproportional wear becomes a major issue. For instance, breadknifing out big chips, or using tape on the spine. Razors were not meant to be chipped, of course, and breadknifing is a logical and practical method of dealing with it. However, it can come with a price, in the form of disproportionate edge wear. Same for habitually taping the spine. The straight razor was not designed with that in mind.

Remember that there is a fairly wide range of bevel angle that will shave, and some steels perform okay with angles slightly smaller or larger than others. I just finished up a Gold Dollar #208 and found the final bevel to be about 18 degrees. The shave is not spectacular, but is pleasantly adequate. Would it shave better at 16 degrees? Probably. But it shaves okay, right now and so there it stands. I have a Universal frameback with a removable blade, which probably originally had a guard, sort of a straight safety razor. I haven't measured it, but my eyeball tells me that the bevel is only about 13 degrees. Way too small. But it shaves, though it is extremely grabby and must be used with care. Would it shave better if I deliberately wore down the edge and increased the bevel to 14 or 15 degrees? Sure. But it shaves now. In both of these examples, examination of the blade geometry offers clues as to why the razor behaves as it does, and while neither one screams out for corrective action, certainly the current angle and its effect on shave quality are to be kept in mind during routine maintenance, with a particular eye toward not making a problem worse. If it aint broke, don't break it, right?

 

[mycarver]

OK,, lets see here. You use the terms such as "if it shaves it shaves" "to tape or not to tape" "wide range of bevel angles that WILL shave" " 13,15,16 degree angles".
Funny,, again in my work they run from 13 to over 20 and they work just fine.
"Meticulous Craftsmen",, "spine softer than the edge"... Do you think the hundreds of razor companies making a wide variety of blades out of whatever stock they had on hand were individually treating the blades as if they were making parts for the Hadron Collider ? Under strict controlled conditions for very specific results there probably IS an ideal angle. And I'm sure as you are shaving you are constantly measuring ,, or have calculated the ideal angle to hold the razor to your face as you go around your cheeks,, your chin, under your neck,,, and have also calculated for the amount of give your skin has to compensate for the changes in angle of attack of the razor to that particular area as to whether it's fleshy skin or backed by bone as this will make a difference as well. And with excellent hand control you can maintain this ideal angle as well.
If you are machining parts angles are important. But with so many variables,, blade width, spine width, steel quality, tempering, angles, ( so much for 'meticulous' craftsmen), angles of the razor to the face, the amount and quality of the lather you aren't machining parts,, you're shaving and you make minute adjustments. Like in carving,, if it isn't cutting,, change the angle of attack,,, these things can be accounted for and simply overcome.
Go back and watch the guy giving the shave with the Kamisori and tell him that with that razor ,, he's using the wrong angles most of the time. It shouldn't be cutting at all and at times he's flat on the guys face and other times nearly 90 degrees to it. All being done with a blade with what ideal angle? And low and behold he actually managed to get a good shave regardless of the angle on the blade. Wow,, I didn't think that was possible. So much for the angle theories I guess because conditions change. Theories are fine in your mind until you have to face the actual use of an item. Then they go out the window . What's the point?

 

[Slash McCoy]

OK,, lets see here. You use the terms such as "if it shaves it shaves" "to tape or not to tape" "wide range of bevel angles that WILL shave" " 13,15,16 degree angles".
Funny,, again in my work they run from 13 to over 20 and they work just fine.
"Meticulous Craftsmen",, "spine softer than the edge"... Do you think the hundreds of razor companies making a wide variety of blades out of whatever stock they had on hand were individually treating the blades as if they were making parts for the Hadron Collider ? Under strict controlled conditions for very specific results there probably IS an ideal angle. And I'm sure as you are shaving you are constantly measuring ,, or have calculated the ideal angle to hold the razor to your face as you go around your cheeks,, your chin, under your neck,,, and have also calculated for the amount of give your skin has to compensate for the changes in angle of attack of the razor to that particular area as to whether it's fleshy skin or backed by bone as this will make a difference as well. And with excellent hand control you can maintain this ideal angle as well.
If you are machining parts angles are important. But with so many variables,, blade width, spine width, steel quality, tempering, angles, ( so much for 'meticulous' craftsmen), angles of the razor to the face, the amount and quality of the lather you aren't machining parts,, you're shaving and you make minute adjustments. Like in carving,, if it isn't cutting,, change the angle of attack,,, these things can be accounted for and simply overcome.
Go back and watch the guy giving the shave with the Kamisori and tell him that with that razor ,, he's using the wrong angles most of the time. It shouldn't be cutting at all and at times he's flat on the guys face and other times nearly 90 degrees to it. All being done with a blade with what ideal angle? And low and behold he actually managed to get a good shave regardless of the angle on the blade. Wow,, I didn't think that was possible. So much for the angle theories I guess because conditions change. Theories are fine in your mind until you have to face the actual use of an item. Then they go out the window . What's the point?

Shave angle is a more individual thing and indeed it is constantly adjusted by the experienced and knowledgeable shaver. Bevel angle is fixed not easily adjusted back and forth, and ideally is, er, within the ideal range. The POINT is to do the best you can do, the best way you can do it, within the scope of practical limitations. Someone who simply HONES a razor, using standard techniques and tools/materials (that includes not automatically taping the spine) will get good results from a good blade, at least most of the time; and if continued for 3 or 4 generations, the same razor will probably continue to serve it's succession of users quite well.

 

[mycarver]

Well there you go,, you made my point.
I know I read it somewhere that it really is that simple ,, except some like to make it more complicated than it really need be.