Looking for a Formula for the Bevel Angle of a Straight Razor

[Lyon Heart]

A few weeks ago, I was exchanging messages on Badger & Blade about my challenges obtaining a decent edge and good shave on a large 11/8ths razor. It was mentioned to me that the angle is likely an issue due to it's size.

I am hoping that amongst the B&B audience there is a mathematician or trigonometrist (sp?) who can tell me if the following is correct. The non-mathematicians can tell me if it's helpful in any way at all.

My thinking is there has got to be a straight forward formula for bevel angle using the ratio of a blade's face size and spine width. This would enable the quick determination of a razors angle as well as the impact of honing with tape.

If a picture tells a thousand words, here is my attempt at a picture:

So I relied on Google a bit, watched a tutorial on trigonometry and the use of inverse sine and applied it to a few of my favorite razors (see below):

Keep in mind that the blade size in the titles are the full blade size but in the table it is the measure from shaving edge to the side of the bevel that would touch the stone when honing.

If this is correct (and please let me know if it isn't) I have three realizations:

First, that there is a huge variation in blade bevel angles. I thought American blades would be narrower but they aren't, they were just smaller. However, I did find some of my older Sheffield razors to be close to 20 degrees, so their comfort isn't simply the difference between wedge vs. hollow ground.

Second, I realized that regardless of whether a razor is 15 degrees or 20, they still shave great if they have a well set edge.

Third, within this range of angle, taping doesn't change it much. Add to this that my measure of taped spines doesn't considered compression and wear during honing. I suspect the impact is half of what the table shows.

I hope that no one has ever been removed from B&B for presenting faulty math.

 

[Steve56]

Without actually checking your numbers, this looks right to me.

Normally, bevel angles run from about 16 - 18 degrees, regardless of the width of the blade. This means that the spine width is varying with blade width. Spine width is also an esy first check if your razor is misbehaving - if tou have a 7/8 - 8/8 with a spine width of 0.200”, you likely need to add tape to get it up to near 0.250” - this is about what most Filarmonica 14s are.

Bevel angle makes a difference, a narrow (lower) bevel angle will shave more aggressively and a wider (higher) bevel angle will shave smoother. For most folks, you’re right, it doesn’t make much difference, but it can be nice to know how to ’tune’ a razor’s angle because not all razors shave the same.

As a rule of thumb, one layer of electrical tape will add about about one degree for widths from 5/8 - 7/8. The wider the blade, the less effect one layer will make, the narrower the blade, the more difference. A layer of electrical tape on a 3/8 will make a difference.

You can have some fun knowing about angles, how to measure them and what bevel angles mean to edge feel on the face.

 

[DrStrange]

A few weeks ago, I was exchanging messages on Badger & Blade about my challenges obtaining a decent edge and good shave on a large 11/8ths razor. It was mentioned to me that the angle is likely an issue due to it's size.

I am hoping that amongst the B&B audience there is a mathematician or trigonometrist (sp?) who can tell me if the following is correct. The non-mathematicians can tell me if it's helpful in any way at all.

My thinking is there has got to be a straight forward formula for bevel angle using the ratio of a blade's face size and spine width. This would enable the quick determination of a razors angle as well as the impact of honing with tape.

If a picture tells a thousand words, here is my attempt at a picture:

View attachment 1710490


So I relied on Google a bit, watched a tutorial on trigonometry and the use of inverse sine and applied it to a few of my favorite razors (see below):

View attachment 1710576

Keep in mind that the blade size in the titles are the full blade size but in the table it is the measure from shaving edge to the side of the bevel that would touch the stone when honing.

If this is correct (and please let me know if it isn't) I have three realizations:

First, that there is a huge variation in blade bevel angles. I thought American blades would be narrower but they aren't, they were just smaller. However, I did find some of my older Sheffield razors to be close to 20 degrees, so their comfort isn't simply the difference between wedge vs. hollow ground.

Second, I realized that regardless of whether a razor is 15 degrees or 20, they still shave great if they have a well set edge.

Third, within this range of angle, taping doesn't change it much. Add to this that my measure of taped spines doesn't considered compression and wear during honing. I suspect the impact is half of what the table shows.

I hope that no one has ever been removed from B&B for presenting faulty math.

I'm getting slightly smaller angles with your data.
Your ratio only has 2 significant figures,
so there can't be more than 2 significant figures in any result based on that.

I do my measurements in millimeters just to make it easier to get more
significant figures.

I don't use the intermediate ratio calculation.

I calculate the bevel angle as
2 * asin (spine / 2 / size)

 

[DrStrange]

Also, I can recognize a 19 degree cutting angle from the way it feels.
Those razors don't make it into my rotation.

Of the razors in my rotation, my 4 favorites
have cutting angles less than 16.5 degrees.
And none of the razors in my rotation
have cutting angles greater than 18 degrees.
I did have some greater than 18
but I can see the trend is that the smaller angles
are going to occupy more of the positions in my rotation.

I rate the quality of razor steel according to how small of an angle
it can support.

 

[Lyon Heart]

I'm getting slightly smaller angles with your data.
......I calculate the bevel angle as
2 * asin (spine / 2 / size)

Thanks for double checking it. Your formula is certainly simpler but just as accurate.

I just added a column with your formula and a second column calculating the difference and got a series of 0.0000. My column was formatted to round the degrees to two digits, so I suspect it was just rounding.

 

[JPO]

Bevel_angle_calculation Johns Razor .xls

bevel angle Bevel angle Calculation Thickness of used tape (in mm):,0,15 Thickness of spine in inches,Width of blade in inches Razor,Thickness of spine,Width of blade,Angle without tape,1 layer,2 layers,3 layers,4 layers Razor 1,5,21,22,2,13,5,14,3,15,0,15,8,16,6,0,2051181102,0,874015748 Razor 2...

docs.google.com

 

[DrStrange]

Also, if I'm taping for angle and not taping to protect the spine,
I hone all the way through the finisher without tape
and then just a little with tape.

 

[Lyon Heart]

Also, if I'm taping for angle and not taping to protect the spine,
I hone all the way through the finisher without tape
and then just a little with tape.

Adding some tape at the end brings me back to some advice that H Brad Boonshaft had for me about adding an extra layer of tape for 4 to 10 strokes on the finishing stone to create a microbevel. In that case, at 15 degrees, the razor was prone to establishing a burr.

This table has helped me understand that the razor I was working with was just over 15 degrees, that my favorite razors tend to be 17 to 18 degrees and that adding a layer of tape added less than a degree to the cutting edge.

My most comfortable razor is an 1820s Joseph Rodgers wedge. I've always assumed old wedges were comfortable due to being wedges, but now I realize the edge (when honed with tape) is over 21 degrees.

 

[JPO]

Adding some tape at the end brings me back to some advice that H Brad Boonshaft had for me about adding an extra layer of tape for 4 to 10 strokes on the finishing stone to create a microbevel. In that case, at 15 degrees, the razor was prone to establishing a burr.

This table has helped me understand that the razor I was working with was just over 15 degrees, that my favorite razors tend to be 17 to 18 degrees and that adding a layer of tape added less than a degree to the cutting edge.

My most comfortable razor is an 1820s Joseph Rodgers wedge. I've always assumed old wedges were comfortable due to being wedges, but now I realize the edge (when honed with tape) is over 21 degrees.

Adding a layer of tape at the end can be really effective and a good solution to a problem that can also be handled differently.
The pressure along the bevel plane is not equal. The bevel is stiffer at the start then behind the apex.
As the edge gets finer and finer it will become more difficult to create a clean apex, because you are not able to generate enough pressure towards the edge without flexing it a little. This creates a small burr.
When you are using something like a jnat with slurry, you can to some degree get around this because the slurry will be effective at the front part of the bevel.
You can also mitigate it by deliberately using a little pressure during the midrange work, and by using light strokes for the final finish.
By using pressure you are deliberately flexing the bevel just enough for you to transfer the pressure forward when you do the finishing.
A third way to do it is use a few minutes during the midrange work on a convex stone. The goal is just to cut away a little steel behind the apex. When you move to the final flat finisher you have moved the pressure slightly forward, which helps you to get a clean edge. It also makes your final stone much faster, because your effective bevel plane area in contact have been reduced.

 

[SliceOfLife]

Spine thickness at any part of spine that has honewear on it = twice opposite.
Distance from the line parallel to the edge where that point resides to the apex of the edge in a straight line = hypotenuse.

Opposite/Hypotenuse = Sin(theta)...where theta is half the edge angle (edge angle on one side of the razor) this assumes the razor is 50/50 edge... not a kamisori. Those also aren't hard to calculate though, just do a measurement for each side and add the resulting theta's directly with the opposite split at the ratio of the grind (70/30 I think Kami's are?).


So lets say you measure spine thickness as 5mm at the very back of the spinewear... And the razor from that point to the edge in a perpendicular line to the edge measures 21mm...

arcsin(2.5/21) gives you the edge angle for that face of the razor to a line bisecting the razor... Multiply by 2 and you have the total edge angle of the razor; 13.67 Degrees.

 

[JPO]

You can also simply the formula if you do not have the trig functions on your calculator.

 

[Bowmaker]

I use a angle protractor or a bevel square with a protractor. It's just easier and faster for me.

 

[Lyon Heart]

I use a angle protractor or a bevel square with a protractor. It's just easier and faster for me.

Actually, I did find that approach to be very straight forward. I used that to confirm the formula. Somehow a formula and result feels more valid (even if the accuracy of the extra digits in the result doesn't make sense given the accuracy of the inputs). Funny thing is, when I did it your way, I didn't feel like I needed a second method to validate the angle.

 

[Dundee]

Degrees Per Side = asin((thickness/2)/width) * 180 / pi

thickness = the lateral thickness of the spine where it will meet the hone
width = the longitudinal distance from the highest point of the spine that will marry to the hone, to the apex of the edge itself

That's just the easy short hand I always used.

So for example one recent blade I restored...

asin((.170/2)/.600) * 180 / pi = 8.144 Degrees Per Side

You can just plug that into Google wherever you are with a smartphone.

You must halve the thickness of the blade to consider it as two right triangles placed back to back, basically as you proposed. However, you just measure the width as being at the very top of where the hone actually touches the spine. You can't, for example, just stick the whole blade between some calipers, because the radius added to the spine will add to this dimension in a way that doesn't actually affect the angle.

I also used this to modify a Gold Dollar razor. It originally came with an angle of about 21 degrees inclusive, but I then filed down the spine width where it would ride on the hone. I took it down to a more acute 16 degrees, but it took a lot longer to reduce the thickness by the necessary amount than you might expect.

asin((.220/2)/.618) * 180 / pi *2 = 20.5 degrees
asin((.165/2)/.618) * 180 / pi *2 = 15.3 degrees

.220 - .165 / 2 = .0275

Might not seem like ~.030 would take to remove off each side, but distributed along the whole spine it's a good bit of metal. But I did it, and it's butt ugly, but it did improve the shave.



Ultimately though, the other thing to realize is that the lateral thickness of the spine will deviate a bit. A tolerance of +/- .005" is achievable by handwork, but much tighter than that usually requires machinery. Keep in mind even the pressure that you push down with can also easily affect things by at least .001" with a very heavy hand.

Anyway, I learned this concept in a machine shop where we often had to calculate an angle with such math. I kind of just think of things a little opposite of you, where 'width' means thickness to me, but otherwise it's basically the same mathematically. The important part to realize is it's all based off the hypotenuse, opposite and adjacent.



In the case or knowing what angle you're grinding at, only the Hypotenuse and Opposite matter for us. The distance from P to Q is the distance from the very top part of the blade to marry the hone surface, to the very edge apex.

So purely in trigonometric terms...

Degrees Per Side= asin(Opposite/Hypotenuse) * 180 / pi

Is the simple formula.

As far as actually verifying this math, from what I have read, using a laser goniometer is best. As the name suggests, it uses a laser to point directly at the blade, and the bevels on the blade will reflect the light off into specific degrees that indicate the bevel angles. To reduce the fanciness, it's basically just laser pointer surrounded by a protractor. Still, they cost $60-$120 to basically accomplish the same thing that math and a cheap set of calipers can measure.

 

[TSENG]

I use this APP to measure the angle of the photo.

 

[SliceOfLife]

I bought some of the "fancier" Gold Dollars a couple years ago and played around with the angles a LOT to determine where they shaved best; how thin I could go before the edge crumbled... How thick before it felt like shaving with a butterknife, etc... Was a very interesting test that only took a few bucks (and an insane amount of low grit honing/grinding).

Basically I found that the steel wasn't quite up to snuff compared to vintages (so if you like 11-15 degree razors... gold dollars/cheap modern razors aren't for you). Can't recall EXACTLY where they survived the shave... but I wanna say it was around 16 degrees minimum to be safe. Thread is around here somewhere.

 

[Dundee]

I bought some of the "fancier" Gold Dollars a couple years ago and played around with the angles a LOT to determine where they shaved best; how thin I could go before the edge crumbled... How thick before it felt like shaving with a butterknife, etc... Was a very interesting test that only took a few bucks (and an insane amount of low grit honing/grinding).

Basically I found that the steel wasn't quite up to snuff compared to vintages (so if you like 11-15 degree razors... gold dollars/cheap modern razors aren't for you). Can't recall EXACTLY where they survived the shave... but I wanna say it was around 16 degrees minimum to be safe. Thread is around here somewhere.

I can believe that. I tried to test my Gold Dollar on the BESS sharpness tester after reprofiling it, and it was insanely high. Turns out the BESS media was denting the blade before it would actually cut through it.

Still, I would rather have to sharpen it more frequently than put up with the 20 degree edge that it came with.

 

[SliceOfLife]

Had one of those myself... damaged every razor even ultra hard steel at high edge angle, but yeah it obliterated the 14-15 degree gold dollars... divots you could see with the naked eye... it was ghastly.

 

[Dundee]

Had one of those myself... damaged every razor even ultra hard steel at high edge angle, but yeah it obliterated the 14-15 degree gold dollars... divots you could see with the naked eye... it was ghastly.

Yeah I have heard a lot of people say none of their straight razors could cut the BESS media without denying. Which is really weird because the people ar BESS use double-edge razor blades as the benchmark for maximum sharpness. They even included razor blades to calibrate with on their prototype model.

Always made me wonder why the DE razors could cut it but not straight razors. Having read that Science of Sharp blog, I think most straight razors rely on an ultra thin wire edge, much thinner than what's on a DE blade. But that still doesn't really explain why straight razors can stand up to hair, but not the plastic media that BESS uses.

The one thing that I remember reading... Though unfortunately don't remember where I read it... Is that the BESS media is case hardened. So the outer layer of plastic is actually harder than the inner core of material. I don't remember the reason for I either, but it could explain why it just crushes straight razor edges.