# Finding pitch in degrees with a rise over run type formula



## LEEQ (Dec 27, 2013)

I saw a posting by Ray C with mention of finding pitch using rise over run . In this method you need to measure over a distance (run) and measuring the difference in pitch(rise) at that point. Having these numbers to input, what do I input them into to work towards an answer in degrees? Can I rearrange the formula to use desired pitch and one other input (rise or run) to give me the other figure? Thanks for any help. I would love to explore this further.


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## John120/240 (Dec 27, 2013)

Machinery's Handbook 23rd Edition has two charts. One is " Rules for figuring tapers" The second table is "Tapers per foot and corresponding angles" Or you could use a section titled "Solutions for triangles"  which uses trigonometry. Google found this-http://www.practicalmachinist.com/vb/general-archive/converting-ft-taper-into-degree-taper-156106/ or this from Starrett http://www.unionmillwright.com/2885.pdf
Hope this helps.


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## LEEQ (Dec 27, 2013)

I'll have to check out that right triangle solution section. I am often overwhelmed with the vast amount of dry reading it takes me to figure if I'm even in the right place with the handbook.  I also forgot to mention that rise and run have a friend named Arctan also at this party.


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## rangerman (Dec 27, 2013)

A "rise" with "run" amount to find the degree of elevation is a simple Tangent formula relationship.

*Tangent of an Angle = rise / run             
*
1 unit of rise for every 1 unit of run is equivalent to the tangent of 45°  (or 100% grade in road construction)


If two of the three unknowns are given you can solve for the third.


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## John Hasler (Dec 27, 2013)

LEEQ said:


> I saw a posting by Ray C with mention of finding pitch using rise over run . In this method you need to measure over a distance (run) and measuring the difference in pitch(rise) at that point. Having these numbers to input, what do I input them into to work towards an answer in degrees? Can I rearrange the formula to use desired pitch and one other input (rise or run) to give me the other figure? Thanks for any help. I would love to explore this further.



The rise divided by the run is the tangent of the angle.  You want to find the inverse tangent (or arctangent) of rise/run.  Thus "angle = arctangent(rise/run)" and "rise/run = tangent(angle)".  Your calculator can figure this for you.  A machinist should know basic trigonometry.  Khan Academy has a course, and there's lots of other stuff on the Web on the subject.


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## Tony Wells (Dec 27, 2013)

Or if you want to cheat, or work some compound angles, there are online calculators, of course.

http://www.pdxtex.com/canoe/compound.htm


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## frank r (Dec 28, 2013)

Divide rise by the run (=the tangent value). Then find that number under the TAN heading on this table:
http://www.industrialpress.com/ext/StaticPages/Handbook/TrigPages/mh_trig.asp#30

Once you find it, the equivalent degrees will be in the left column under Angle.

Notice the source of this table is the Machinery's Handbook.


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## LEEQ (Dec 28, 2013)

So, no less confused. It seems as if some are saying I want tan and some arctan. I might need both. What I am hoping to accomplish is to learn how Ray C ran the calculations. I'm dreaming I can learn the formula and not need to use charts. I prefer to be able to scratch math out with pencil and paper. I then leave myself the process and examples in my notes.  I can refer to these and remember how to figure things such as how to set up my rotary table for dividing. If I can get this process down, I can use it to do different angle set ups. I learn best through interaction and was hoping someone could lead me through this in terms vastly more simple than proper technical math terms. Explaining the proper terms so a simpleton( ie me) gets it would be great too.           " the spindle DI gave me 0.025" the carriage moved 0.200". Arctan (0.025/0.200) = 7.125*." This shows the Problem as Ray layed it out.


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## Ray C (Dec 28, 2013)

If there's some other example you'd like to review, let me know.  If you were measuring the angle of the lip on the D1 spindle, it seems you got it right.  The edge of the lip is not very wide so that's probably how much the needle deflected downward or upward when you moved the DI 0.2" horizontally.  Therefore the rise (or drop depending on which way you were moving) was 0.025 and run was 0.200".    Arctan (0.025/0.200) = 7.125*.   That by the way, is the one-side angle.  The included angle (the angle from the centerline) is 14.25* which takes into account both angles.

Now, lets turn it around...  What happens if you only know the one-side angle and someone tells you it's 7.125*.  What would be the rise over run?

EDIT:  In the original post here, I mis-wrote something.  What follows has been edited/corrected as not to cause anyone confusion. -ray


With a calculator, calculate Tangent (7.125).  It equals:  0.125.   Now, you just find any two numbers whose quotient equals 0.125.  That is A / B = 0.125.   Pick a number, any number...  Let's say 3.  Now substitute 3 for either A or B; I'll use A.

3 /  B = 0.125".   or, B = 3 x 0.125 which equals: 0.375.   This means that if you move in 3" in the horizontal direction, the DI needle should deflect 0.375". 

The trick of course, is to pick the first number in a range corresponding to the distances you're working with.

Does that help?

Ray




LEEQ said:


> So, no less confused. It seems as if some are saying I want tan and some arctan. I might need both. What I am hoping to accomplish is to learn how Ray C ran the calculations. I'm dreaming I can learn the formula and not need to use charts. I prefer to be able to scratch math out with pencil and paper. I then leave myself the process and examples in my notes.  I can refer to these and remember how to figure things such as how to set up my rotary table for dividing. If I can get this process down, I can use it to do different angle set ups. I learn best through interaction and was hoping someone could lead me through this in terms vastly more simple than proper technical math terms. Explaining the proper terms so a simpleton( ie me) gets it would be great too.           " the spindle DI gave me 0.025" the carriage moved 0.200". Arctan (0.025/0.200) = 7.125*." This shows the Problem as Ray layed it out.


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## frank r (Dec 28, 2013)

You need both. It is not easy to do by hand. John Hasler's response above is correct. You need to use a calculator with an arctan (AKA: inverse tan) function to calculate them.

I just finished a pre-calculus course and we studied this topic. We also had to use a graphing calculator (about $100) to figure out our answers. If you don't have the calculator you can use the table.


tan 45 degrees = 1 (that is a rise of 1 divided by a run of 1) [the tangent will result in a number]

The arctan of 1 = 45 degrees [the arctan will result in a degree]


Note that inverse tan does not mean the reciprocal of tan (you can't just flip the rise over run and get the correct answer).


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## rangerman (Dec 28, 2013)

Someone already suggested it, but here's a good online lesson on Basic Trigonometry.

https://www.khanacademy.org/math/tr...ometry/basic_trig_ratios/v/basic-trigonometry

Everything is explained at the beginners level like it should at least have been taught in every High School math class.

It always makes it easier for one to draw his triangle first in order to have a better visual idea of the relevant dimensions involved in relation to his actual problem at hand.

You'd still get lost in solving a machining problem like a taper degree determination if you couldn't identify which of the triangle sides refer to which in the actual part being machined.


Note: The determination of the correct ratios between triangle sides that would correspond to a certain value in degrees requires a* Trigonometry table or a Scientific Calculator*. 
  The actual calculation involved is beyond the basics of trigonometry.


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## LEEQ (Dec 28, 2013)

Ray C said:


> If there's some other example you'd like to review, let me know.  If you were measuring the angle of the lip on the D1 spindle, it seems you got it right.  The edge of the lip is not very wide so that's probably how much the needle deflected downward or upward when you moved the DI 0.2" horizontally.  Therefore the rise (or drop depending on which way you were moving) was 0.025 and run was 0.200".    Arctan (0.025/0.200) = 7.125*.   That by the way, is the one-side angle.  The included angle (the angle from the centerline) is 14.25* which takes into account both angles.
> 
> Now, lets turn it around...  What happens if you only know the one-side angle and someone tells you it's 7.125*.  What would be the rise over run?
> 
> ...



I did the work in parentheses first. rise overrun which gives me 0.125. That is the tangent. That is as far as I know how to get. How do I get from rise over run to pitch in degrees? I can get a scientific calculator from the kids. Lord knows I've bought enough of them. I also have 23rd edition M H.


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## frank r (Dec 28, 2013)

LEEQ said:


> I did the work in parentheses first. rise overrun which gives me 0.125. That is the tangent. That is as far as I know how to get. How do I get from rise over run to pitch in degrees? I can get a scientific calculator from the kids. Lord knows I've bought enough of them. I also have 23rd edition M H.




Refer to my first post. Its all there. Once you have the tan, you use a calculator or tables to find the arctan (which is 7.125 degrees).


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## Ray C (Dec 28, 2013)

Once you get a scientific calculator pryed away from one of the kids, enter 0.125 then, find the arctan button.  It may be shown as ATAN, Arctan or TAN[SUP]-1.

[/SUP]As someone else mentioned, Tan[SUP]-1[/SUP] does not mean the reciprocal of the number, it's just a different way of writing arctan.  When you press the arctan button, that should give you 7.125.  On some calculators, you may need to press a shift key to get the arctan function.


 Ray

PS.  I accidentally hit the Edit button of your post when I meant to press "Reply with Quote".  As a moderator, I'm able to edit other people's posts.  It was an accident and I did not change anything in your post.





LEEQ said:


> I did the work in parentheses first. rise overrun which gives me 0.125. That is the tangent. That is as far as I know how to get. How do I get from rise over run to pitch in degrees? I can get a scientific calculator from the kids. Lord knows I've bought enough of them. I also have 23rd edition M H.


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## LEEQ (Dec 28, 2013)

frank r said:


> Refer to my first post. Its all there. Once you have the tan, you use a calculator or tables to find the arctan (which is 7.125 degrees).


how would I use a calculator instead of tables? how do I do the work?


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## Ray C (Dec 28, 2013)

See Post #14.  Also, some calculators call it INVTAN (meaning, inverse tangent).

Ray





LEEQ said:


> how would I use a calculator instead of tables? how do I do the work?


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## LEEQ (Dec 28, 2013)

Ray C said:


> Once you get a scientific calculator pryed away from one of the kids, enter 0.125 then, find the arctan button.  It may be shown as ATAN, Arctan or TAN[SUP]-1.
> 
> [/SUP]As someone else mentioned, Tan[SUP]-1[/SUP] does not mean the reciprocal of the number, it's just a different way of writing arctan.  When you press the arctan button, that should give you 7.125.  On some calculators, you may need to press a shift key to get the arctan function.
> 
> ...


Ok, just find tangent. rise over run = tan. Then hit arctan to get degrees.
Ok, So if I enter the number 7.125 and hit tangent it will spit out 0.125? Then if I move the table over .500, I need to move it .250 in to be at the7.125 angle? 
as you said A x B =  tan. A being .500, divide tan .125 by A getting .250 for B. I think that gets me through the problem without charts. I need a good calculator and follow everything lined out right here in this post between the two of us, right?


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## Ray C (Dec 28, 2013)

I'm sorry but I made a little error before but, you have the right idea.  Here is the correction.

Tan(7.125) is 0.125.

Rise/Run = 0.125 therefore, Rise = 0.125 x Run.   If you want the Run to be 0.500 then, Rise will be 0.0625.

Proof:

0.0625/0.500 = 0.125  It works.

Also, on your calculator, if you take Arctan(0.125), you will get 7.125

... I'm sorry, in my earlier post I said the product of two numbers and I meant the quotient which is the result of dividing two numbers.  Truth be know, I've had a splitting headache all darn day and my head was not in the right place.

Ray











LEEQ said:


> Ok, just find tangent. rise over run = tan. Then hit arctan to get degrees.
> Ok, So if I enter the number 7.125 and hit tangent it will spit out 0.125? Then if I move the table over .500, I need to move it .250 in to be at the7.125 angle?
> as you said A x B =  tan. A being .500, divide tan .125 by A getting .250 for B. I think that gets me through the problem without charts. I need a good calculator and follow everything lined out right here in this post between the two of us, right?


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## LEEQ (Dec 29, 2013)

Ray C said:


> I'm sorry but I made a little error before but, you have the right idea.  Here is the correction.
> 
> Tan(7.125) is 0.125.
> 
> ...


I think you did just awesome. I'm all straightened out with simple useful knowledge I can refer back to without taking college level math. That's just great. I appreciate everyone's help, but you made it simple enough for even me. I hope others get to benefit from the outcome of this thread, thanks.


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## lnr729 (Dec 29, 2013)

LEEQ said:


> I'll have to check out that right triangle solution section. I am often overwhelmed with the vast amount of dry reading it takes me to figure if I'm even in the right place with the handbook.  I also forgot to mention that rise and run have a friend named Arctan also at this party.



I wrote a program for calculating solutions to triangles and a few other useful things around the shop.

http://www.dogcreek.ca/shopcalc/WorkshopCalulator.html


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## astjp2 (Dec 29, 2013)

Speaking of pitch, how would you figure out the pitch of a worm gear based on a screw pitch?  For example, a 3/4x8 acme screw, what pitch is the worm gear for it?  14.5* pressure angle.  Tim


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## frbutts (Dec 30, 2013)

1 deg. equals .0174"per inch so run times 1 divided by .0174 equals degrees


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## rangerman (Dec 30, 2013)

frbutts said:


> 1 deg. equals .0174"per inch so run times 1 divided by .0174 equals degrees



That's an approximation adequate enough for very small angles of a few degrees, but it would create increasingly larger errors for angles greater than 10 degrees.
Or, to put it another way, .0174" rise for every 1" run is only mathematically exact for 1 degree angle.

Example: Tan 45° = 1  =  [ .0174" rise  over .0174" run ] , but it is not equal to .0174 x 45


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