# How do I determine a taper angle, given length & width at both ends?



## cazclocker

I'm fabricating a replacement for the part shown in my attached sketch. The severely-worn part is my locking cam lever, used to secure the tailstock of my watchmaker's lathe to the lathe bed ways. I've got the main body made (pretty simple, really) done - now it's time to turn a new handle.

So...my question is, what is the formula to figure the half-angle for the tapered part (obviously, minus the ball-end)? The length is 0.758" (but I'll call it 0.760" for simplicity's sake), and the diameters are 0.146" at the narrow end, and 0.218" at the larger end.

Instead of you guys just feeding me the answer (although that would be nice, too, just so I can get going on the project!), I would like to know what the correct formula is, so I can figure similar equations in the future.

Thanks!
...Doug


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## Ray C

You will need to use the Arctan function on a calculator or look it up in tables. Sometimes the button is called ATAN or TAN[SUP]-1

[/SUP]Anyhow, the formula is Arctan (change in height / distance).

In your case this will be Arctan [ (0.218 - 0.146) / .758 ]. This is 5.43 degrees. Since you are measuring diameters, this is the full angle so we still need to divide by 2 to get the half angle you requested thus, it's 2.72 degrees.


Ray

EDIT:  On some calculators Arctan can also be called Invtan or TanInv. The number sequence on a typical calculator is like this:

0.218
-
0.146
=
(the answer will show up as .072)
/
.758
=
(the answer will show up as 0.095)
ArcTan
(the answer will show up as 5.4261....)


PS:  Please don't be offended by the explanation of the calculator.  A long time ago, I had a post which needed the calculator and I received many PMs on how to enter the numbers so, every now and then, I cover the material again.


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## cazclocker

Ray C said:


> You will need to use the Arctan function on a calculator or look it up in tables.  Sometimes the button is called ATAN or TAN[SUP]-1
> 
> [/SUP]Anyhow, the formula is Arctan (change in height / distance).
> 
> In your case this will be Arctan [ (0.218 - 0.146) / .758 ].  This is 5.43 degrees.  Since you are measuring diameters, this is the full angle so we still need to divide by 2 to get the half angle you requested thus, it's 2.72 degrees.
> 
> 
> Ray



Thank you, Ray. I am not strong in trig at all, but I have a pretty good scientific calculator with a TAN(superscript -1) button, which I guess is the function you're referring to. I still don't know how to apply the formula you supplied to the buttons on my calculator, but fortunately I still have the manual! So hopefully a little studying up will be in order...
...Doug


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## Ray C

I edited the thread to provide some calculator instructions.  Have a look again.  Or, you can tell me the type of calculator and I can walk you thru.


Ray




cazclocker said:


> Thank you, Ray. I am not strong in trig at all, but I have a pretty good scientific calculator with a TAN(superscript -1) button, which I guess is the function you're referring to. I still don't know how to apply the formula you supplied to the buttons on my calculator, but fortunately I still have the manual! So hopefully a little studying up will be in order...
> ...Doug


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## cazclocker

Ray C said:


> I edited the thread to provide some calculator instructions.  Have a look again.  Or, you can tell me the type of calculator and I can walk you thru.
> 
> 
> Ray



Hi Ray! I followed your revised instruction and it works great! Thanks! Good grief, I am not offended at all - quite the contrary, what I needed was for someone to literally grasp my index finger and make me hit the right buttons. Short of that, I can follow instructions very well, and yours are very clear...step-by-step! My calculator is a pretty old solar-powered Casio fx-85, and the arctangent button is marked with "tan(superscript -1)" .... by the way, how do you get the actual little "-1" miniaturized and up in the air like you do? Anyway, my arctangent button doubles as a tangent button, so I have to hit the INV button to make the tangent button do the arctangent function.

The important thing is, I now know how to arrive at the same result that you supplied, so I must be doing it right. So now, if I want a slightly different angle all I have to do is duplicate all the moves but substitute different numbers.

Thank you Ray!!
...Doug


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## Ray C

Glad the slope/taper problem had a happy ending and properous looking future! And by the way, that same formula is used for any slope problem such as for calculating the slope of a staircase, hill, mountain, wheelchair ramp... Just gotta watch out if you need the half-angle. For a wheelchair slope you are measuring the heights from a flat centerline so, just subtract the heights and divide by distance and hit arctan. No need to divide the answer by 2 in that case. Easy Spheasy...


When I press the reply button, I get 3 bars of editing options just above the editing window. On the 3rd row, there is X[SUB]2[/SUB] and X[SUP]2[/SUP] for subscript and superscript.

So... if you want to write Tan-1 just write it then, highlight just the "-1" and press the sub or super-script button. Now watch-out for something. You need to leave a space character after highlighting the "-1". If you don't everthing you type afterwared will show-up as scripted.

Ray

PS: I get the biggest kick out of your icon picture. There's another one like that with a kid sticking a screwdriver in a socket.





cazclocker said:


> Hi Ray! I followed your revised instruction and it works great! Thanks! Good grief, I am not offended at all - quite the contrary, what I needed was for someone to literally grasp my index finger and make me hit the right buttons. Short of that, I can follow instructions very well, and yours are very clear...step-by-step! My calculator is a pretty old solar-powered Casio fx-85, and the arctangent button is marked with "tan(superscript -1)" .... by the way, how do you get the actual little "-1" miniaturized and up in the air like you do? Anyway, my arctangent button doubles as a tangent button, so I have to hit the INV button to make the tangent button do the arctangent function.
> 
> The important thing is, I now know how to arrive at the same result that you supplied, so I must be doing it right. So now, if I want a slightly different angle all I have to do is duplicate all the moves but substitute different numbers.
> 
> Thank you Ray!!
> ...Doug


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## Ray C

... And we all make mistakes -and I failed to mention a longer (but more precise) method of calculating angles in the case of the original post.

Sit tight, I need to draw some pictures to show things properly.  Don't despair, the quick method is generally correct in the 1st and 2nd decimal place and perfectly suited for things of a "ornamental" nature.  BTW:  An angle represented out to one decimal place is generally hard to machine and two decimal places is ten times harder.  Three decimal places?  Probably ain't going to happen on homeshop equipment...

I'd like to thank Terry for reminding me offline that I should show both techniques for the sake of completeness...


Ray


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## easymike29

It's important to recognize the difference between these two examples of taper. It's not a good idea to think that all tapers look enough alike to think that one formula fits all.

Gene


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## Ray C

Correct.  For the figure on the left you'd use ArcSin instead of ArcTan.

However, this is not where the additional complexity comes into play. I'm making some drawings to illustrate...


Ray




easymike29 said:


> It's important to recognize the difference between these two examples of taper. It's not a good idea to think that all tapers look enough alike to think that one formula fits all.
> 
> Gene


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## Ray C

OK, here's "the rest of the story"...

Please take a look at this example. The geometries are chosen to really show the problem in the worst case.(You can download the file to see a better version).


If you do the quick & dirty method, you simply calculate Arctan (8-0.5)/15 as 26.565. The half angle is 13.2825.
The correct (most accurate) method is is to subtract the differences of the halves which is Arctan (4 - 0.25)/15 = 14.0362. This time, we do not divide by 2. There is a difference of 0.754 degrees in the answers.

If you calculate the values from the original post using the precise method, the final answer comes to 2.7191. The shortcut method showed an answer of 2.72.

This is kinda my bad because when I glanced at the OPs original measurements, I instinctively knew the results would be about the same...

Ray

EDIT: I've included a larger version of the same file. Hope it's bigger this time.

View attachment document.jpg


View attachment document.jpg


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## cazclocker

Ray C said:


> Glad the slope/taper problem had a happy ending and properous looking future! And by the way, that same formula is used for any slope problem such as for calculating the slope of a staircase, hill, mountain, wheelchair ramp... Just gotta watch out if you need the half-angle. For a wheelchair slope you are measuring the heights from a flat centerline so, just subtract the heights and divide by distance and hit arctan. No need to divide the answer by 2 in that case. Easy Spheasy...
> 
> 
> When I press the reply button, I get 3 bars of editing options just above the editing window. On the 3rd row, there is X[SUB]2[/SUB] and X[SUP]2[/SUP] for subscript and superscript.
> 
> So... if you want to write Tan-1 just write it then, highlight just the "-1" and press the sub or super-script button. Now watch-out for something. You need to leave a space character after highlighting the "-1". If you don't everthing you type afterwared will show-up as scripted.
> 
> Ray
> 
> PS: I get the biggest kick out of your icon picture. There's another one like that with a kid sticking a screwdriver in a socket.



OK, I get it now!! I also read the subsequent posts about different kinds of angles...so now I know not 1 equation solves them all. It's nice to know how to use the TAN[SUP]-1[/SUP] button (hey look everybody...I can make superscripts!!)
Thanks for the comment re: my avatar...I chose it because for some reason I can't use an animated .gif file as an avatar. On my clock repair message board I belong to, I use THIS for an avatar....


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## easymike29

You will never have trouble with angles if you understand this. Keep a copy of it with you if you can't memorize it.

Gene


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## cazclocker

easymike29 said:


> You will never have trouble with angles if you understand this. Keep a copy of it with you if you can't memorize it.
> 
> Gene



Gene, you're a gentleman and a scholar! There's no way I could memorize that - so I'll print it out on card stock and give it to my buddy who has a laminating machine. That one is a KEEPER!
Thanks so much, all you helpful and more experienced guys...
...Doug


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## Ray C

FYI:  You're always going to use the same formula that was provided but, the numbers for the "change in height" are derived a little differently.  If you have a part with an extreme taper (goes from very small to very wide over a short distance) the approximate method does not give as good a result.  With very minor tapers, the differences in the methods only show-up at or beyond the second or third decimal point.  I don't think it's possible to cut tapers in any homeshop machine with accuracy at or beyond two decimal points.  When I cut B&S tapers it's darn near impossible to differentiate between say 7.13 vs the desired 7.125.  The resulting hubs I make still run true within +/- 0.0002" and there is no perceptible wiggle when the shaft is inserted in the taper hole.  A fleck of dust or heavy smear of grease will throw you off as much or more.

Ray




cazclocker said:


> OK, I get it now!! I also read the subsequent posts about different kinds of angles...so now I know not 1 equation solves them all. It's nice to know how to use the TAN[SUP]-1[/SUP] button (hey look everybody...I can make superscripts!!)
> Thanks for the comment re: my avatar...I chose it because for some reason I can't use an animated .gif file as an avatar. On my clock repair message board I belong to, I use THIS for an avatar....


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## Jimbo

Doing the math and getting the angle is good, but then you have to find a way to set the compound slide to that exact angle.  If you can remove the handle and chuck it up straight in lathe then you can put a dial indicator in the tool holder and adjust to make a perfect match if it is critical.


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## mekanix48

Hi Jimbo.
For what it's worth here's a small contribution to help you out.. if it displays ok, otherwise it's a scan from the trig' page in the Zeus pocket book.




Regards
George


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