Sharpening freehand on a benchstone - formula

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rayback

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Sep 16, 2006
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In Leonard Lee's book - The Complete Guide To Sharpening, Lee shows a way to put a 5 degrees back-bevel on a tool.
He calls it The 1-In-60 Rule.
I've thought about a way to elaborate and translate his words into an easy math formula which will help us put whatever angle we want on our tools when shaprening freehand on a benchstone.
The formula I'm about to present will help us determine the height we need to hold the knife from the benchstone in order to achieve a certain angle.

I will use for that a right triangle which exactly resembles the job of sharpening a knife.

Here is the right triangle:
viewphoto


According to the picture, the hypotenuse of the triangle equals to the knife's blade width, the angle 'a' is the sharpening angle which is created between the knife blade and the benchstone, and 'H' is the height we need to raise up the knife above the benchstone.
In order to determine the H - height for a certain angle we will use the following formula (Trigonometry):

Sin a = Opposite normal/Hypotenuse = H/Hypotenuse

Let's for example determine the height for a 2 inches wide knife, 15 degress sharpening angle.

Sin 15 = H/2
H = Sin 15 X 2 = 0.51inches

We have to raise the knife 0.5 inches above the benchstone.

Let's conclude the formula:

H = Sin a X Knife-Width
a = target angle


Hope it will be useful for you guys
-Ray

p.s. I couldn't manage to upload the pic, If anyone has here a free server It will help, I'm gonna send him the pic and I'll update the link.
 
tell me how did you get 15 to begin with? if i have a folder with a 1 inch blade aus 8 steel and i want to freehand sharpen with a stone. what angle would i begin with?
 
You have to put into the formula whatever sharpening angle you desire.

For a 1" wide blade, and a 15 degrees sharpening angle:
H= sin 15 X 1 = 0.25 inches.
It means you will have to hold the knife in a way, which the height from the blade to the benchstone would be 0.25inches.
 
Cool thanks for the info. I never know what my knife bevels are at because i've never measured them. However I may just start chacking them after I sharpen them. I just sharpen the blade to the desired task at hand and chop or slice away. I'm one of those guys who like the thinest edges possible on my knives without excessive chipping or rolling, but it would be nice to see what those angles really are and with simple math even I can figure that out.;)
 
You welcome!
I'm glad I can be of some help.
If anyone needs some clarification about anything please ask.
English isn't my native language, so I hope you've guys understood the math and the explanations.
 
Lee's forumla is usually used by tradesmen, it is based on the approximation that sin a = a if a is a small angle (and in radians). This is then convered to degrees as 1 radian is approximately 60 edgrees. When measuring the heights take care to do so from the middle of the spine.

-Cliff
 
Putting it all together, I think the formula below works well for angles less than 45 degrees.

Spine height as fraction of blade width = (Angle in degrees) / 60

Required spine height:
for a 30 degree angle = half the blade width
for a 20 degree angle = third of the blade width
for a 15 degree angle = fourth of the blade width
for a 10 degree angle = sixth of the blade width

Maybe some day I'll try freehand again.:rolleyes: It always went badly when I tried before. :o
 
wow...confusing.

I've been sharpening so long that I can just look at the blade, the original edge bevel, use my eyes and hands and sharpen freehand without worrying about all this angle talk. I think this kind of stuff is what discourages people from sharpening freehand on a stone IMO.
 
Cliff Stamp said:
Lee's forumla is usually used by tradesmen, it is based on the approximation that sin a = a if a is a small angle (and in radians). This is then convered to degrees as 1 radian is approximately 60 edgrees. When measuring the heights take care to do so from the middle of the spine.

-Cliff
Click to expand...

1 radian = 57 degrees.
There is a law in physics which says that if we use lower the 20 degrees angles we can still use the approximate way and still be very close.
Lee bases on this law.
The formula I gave is a more accurate way to do so.
As long as the degrees are less than 20, we can use the 1-60 rule, otherwise the result will lose its accuracy.

In the formula I wrote the height of the knife should be measured from one end to the other.
The hypotenuse is actually the knife being held in a slope creating an angle between the base of the triangle (which is the benchstone in real life). In the picture I've called it "Knife width"... I meant to the Knife height I guess.
 
also agree with pysco78. sharpening doesn't have to be hi tech. I sharpen the same way you do and it works great for me.
 
rayback said:
1 radian = 57 degrees.
Click to expand...

A radian is defined by 2pi radians in a unit circle, so one radian by defination is 180/pi degrees which is indeed approximately 57 degrees. This is usually approximated further to 60 by tradesmen because the error in doing so is less than 5% and few tradesmen carry tools which can measure to that extent. 60 is also a lot easier to multiply than 57.

There is a law in physics which says that if we use lower the 20 degrees angles we can still use the approximate way and still be very close.
Click to expand...

The approximation is based on a truncation of the taylor series expansion of sin around zero, one term retained gives sin a = a. In regards to laws, there is no law in physics about when to truncate a series though every individual can set rules and guidelines. For series approximations you look at the remainder term which in this case is x^3/3!. Thus the maximum percentage error in using this approximation would be :

((angle in degrees)/180*pi)^2/3!

Knife angles are all very acute in general and thus this remainder term is quite small and thus you only need one term in the sin series. Even if you were using very precise jigs you would not need more terms at common sharpening angles but in some cases you would not want to use the coarse 60 degrees = one radian approximation because the uncertainty there is about 5% and some people do set microbevels at smaller increments.

-Cliff
 
Cliff, I guess I completely misunderstood Lee's theory.
So if I understand you correctly he uses Radians in the 1-In-60 Rule.
You taught me something new...
What I couldn't also understand is why do we have to measure the height of the knife from the middle of the spine in Lee's theory?

Thanks
Ray
 
rayback said:
So if I understand you correctly he uses Radians in the 1-In-60 Rule.
Click to expand...

The one in sixty rule comes from the conversion of radians to degrees after the approximation that sin a = a. If you worked in radians it would be a 1 - in - 1 rule.

why do we have to measure the height of the knife from the middle of the spine in Lee's theory?
Click to expand...

That is how angles are measured in general. For example when using a Sharpmaker you keep the spine perpendicular to the floor and thus the angles are set in regards to the midpoint line of the knife. Steve Bottorff talks about both angle measurements in "Sharpening made Easy."


-Cliff
 
Couple thoughts .... If you're technology-enabled, which everyone here obviously is, it seems just as easy to find the sine of the desired angle using an online conversion, spreadsheet, calculator, etc. Multiply the result by the blade width and you're set.

The bigger problem for most IMO is actually holding that angle, especially when it's reasonably acute and the blade not very wide. For example take a .7" wide blade: to sharpen at 10 deg/side you want to have the center of the spine .12" off the stone; let the spine rise to .15" while sharpening and you're now >12 deg/side. It's easy to slip up 3/100ths of an inch or more when working with a stone laying flat on a bench or held in the hand if you're just using visual cues or even a thumb or fingers as a guide. So as it turns out both the 1-in-60 rule and actual trigonometry are far more accurate than the application requires. As pysco78 says it's probably easier for most to just concentrate on matching the existing bevel.

All of which leads me to conclude that when sharpening freehand it's much more practical to not worry about accuracy when cutting relief, just make sure it's substantially more acute than you want the final edge. Then apply a secondary bevel and/or microbevel to give you the final edge geometry you want, based upon how you intend to use the knife.
 
I find that most of the time I can use my thumb as a guide on the bench stone, tho I suspect the angle migrates over time. I saw this in the NYT about sharpening knives using pennies to measure the distance between the stone and the knife. Raise it one penny or two etc. Can remind the thumb of the position. This probably requires a subscription to view, but here it is:
http://select.nytimes.com/search/restricted/article?res=FB0D1EFC35550C708EDDA00894DE404482
 
Dog of War said:
All of which leads me to conclude that when sharpening freehand it's much more practical to not worry about accuracy when cutting relief, just make sure it's substantially more acute than you want the final edge.
Click to expand...


Yes, that is what Lee recommends but you should be a little more focused as it is ground to the same actual constraint as the edge (minimal cross section for durability) but in most cases outside of the extreme (heavy tactical) that is often reduced to the lowest you care to grind. I just grind them all basically flat to the blade, even for large chopping knives which get only a few degrees of "roll" as Fikes demonstrates, as the primary grind is usually about 5 and I want the relief grind for the edge to be about 8-10 at maximum.


-Cliff
 
My hand naturally holds a blade at about 10-13 degrees to the stone, I can also follow a stock grind without much problem, but I don't like to. I do find that when I am pulling the edge toward myself, I seem to let the spine drift lower at times. Can be a hassle when trying to keep the blade finish nice and even.

There really isn't anything confusing or hi tech about this. If you can hold a consistent angle, then all you need do is measure it once and perform the simple calculation. Measuring this angle isn't difficult, holding it is, which is why I believe there are people who don't want to sharpen their knives. That, and bad experiences with poor knives.
 
hardheart said:
Measuring this angle isn't difficult, holding it is, which is why I believe there are people who don't want to sharpen their knives. That, and bad experiences with poor knives.
Click to expand...

Most people are satisfied if their knives can cut at all. I have given people knives that were just shaped while I was working through a set and they were still quite pleased. The biggest problem is that they start off with stones which are much too fine and thus spend a long time doing nothing but regrinding the shoulder. If people started off with x-coarse silicon carbide then few people would complain about how difficult it is to sharpen a knife, aside from those that wanted to generate an edge which would satisfy Clark.


-Cliff
 
The biggest problem is that they start off with stones which are much too fine and thus spend a long time doing nothing but regrinding the shoulder. If people started off with x-coarse silicon carbide then few people would complain about how difficult it is to sharpen a knife, aside from those that wanted to generate an edge which would satisfy Clark.
Click to expand...
So true! I think this is the problem for beginners most of the time. Then when they give up they assume they need an even finer stone.
 
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