# Calculate angle from drawing



## RichO (Apr 29, 2009)

At the cabinet shop I work at, we build compound mitered wood hoods for many homes. While there are different ways to build one, the guy that heads my department draws it out full scale on a sheet of wood and then uses a digital protractor to get the correct angle to cut the pieces at.

I am not much of a geometry guy but there must be some kind of formula you can use to get the angle based on the measurements, like in the attached drawing.

I searched google for finding the angles in a trapezoid but the only results I am getting are based on the other angles, not the measurements. Looking to find the angle at the red arrow.

Thanks for the help.


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## Dave McCann (Jun 21, 2020)

The angle in question is 77.9 degrees for the example shown. Keeping in mind that the information given is only enough to calculate the simple angle shown, not a compound angle. To figure the angles as in the example, simply "remove" the 24 by 28 rectangle in the center and trig out the angles of each right angle triangle on each end.


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## RichO (Apr 29, 2009)

Well, "trig out" is a term that is over my head LOL


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## Dave McCann (Jun 21, 2020)

RichO said:


> Well, "trig out" is a term that is over my head LOL


"Trigonometry," is the basis for all calculations of a right triangle. Something which will serve you well if you will be working with simple and compound angles.

Right triangle
A right triangle is a type of triangle that has one angle that measures 90°. Right triangles, and the relationships between their sides and angles, are the basis of trigonometry.
In a right triangle, the side that is opposite of the 90° angle is the longest side of the triangle, and is called the hypotenuse. The sides of a right triangle are commonly referred to with the variables a, b, and c, where c is the hypotenuse and a and b are the lengths of the shorter sides. Their angles are also typically referred to using the capitalized letter corresponding to the side length: angle A for side a, angle B for side b, and angle C (for a right triangle this will be 90°) for side c


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## woodnthings (Jan 24, 2009)

*When trig is too much .....*

You can always make a layout on your bench top and just measure the angle with a digital angle finder or a protractor:












In this case, subtract 24 from 36 and get 12. Strike a vertical line using a drywall square from the edge of your bench and make a tick mark at 28" up.

Move over 6" at the base (because 6 is 1/2 of 12 and you have 2 sides or angled pieces), and make a tick mark.
Connect the marks.
That's your angle.
Set the protractor on the base or bottom edge and read the angle.
No math required. :surprise2:


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## Dave McCann (Jun 21, 2020)

woodnthings said:


> You can always make a layout on your bench top and just measure the angle with a digital angle finder or a protractor:
> 
> No math required. :surprise2:


Yep that what he's doing now, yet he asked about the math.


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## Echo415 (Apr 3, 2018)

You need to find the angles in order for the formulas to work but yes they do exist...find a calculator that can do both sin/cos/tan as well as their inverses. I'll post the formulas for compound miters later when I'm less sleep deprived. Remember sin = opp/hyp, cos = adj/hyp, tan = opp/adj.


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## RichO (Apr 29, 2009)

Dave McCann said:


> "Trigonometry," is the basis for all calculations of a right triangle. Something which will serve you well if you will be working with simple and compound angles.
> 
> Right triangle
> A right triangle is a type of triangle that has one angle that measures 90°. Right triangles, and the relationships between their sides and angles, are the basis of trigonometry.
> In a right triangle, the side that is opposite of the 90° angle is the longest side of the triangle, and is called the hypotenuse. The sides of a right triangle are commonly referred to with the variables a, b, and c, where c is the hypotenuse and a and b are the lengths of the shorter sides. Their angles are also typically referred to using the capitalized letter corresponding to the side length: angle A for side a, angle B for side b, and angle C (for a right triangle this will be 90°) for side c


I do understand that part of it. It becomes a triangle with one 6" side, one 28" side and one 90 degree angle. Using the Pythagorean theorem, the hypotenuse is 28.635". So, using those figures, how did you come up with 77.9 degrees for the angle?


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## RichO (Apr 29, 2009)

Dave McCann said:


> Yep that what he's doing now, yet he asked about the math.


Yes, just figured a few numbers punched into a calculator would be a quicker method. Correct me if I am wrong


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## kwoodhands (May 1, 2020)

I would draw a triangle that is 28" high , 6" at the base. Connect the two and this gives you the angle needed.
6" is half the distance from 36"-24"=12", divide in half and it is 6". Transfer the angle to a bevel square. Then to your saw. I made a large bevel square from scrap maple that is 18" long . This would make the angle set more accurate .
mike


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## Dave McCann (Jun 21, 2020)

RichO said:


> I do understand that part of it. It becomes a triangle with one 6" side, one 28" side and one 90 degree angle. Using the Pythagorean theorem, the hypotenuse is 28.635". So, using those figures, how did you come up with 77.9 degrees for the angle?


Basic Formulas.
sin θ = Opposite Side/Hypotenuse.
sec θ = Hypotenuse/Adjacent Side.
cos θ = Adjacent Side/Hypotenuse.
tan θ = Opposite Side/Adjacent Side.
cosec θ = Hypotenuse/Opposite Side.
cot θ = Adjacent Side/Opposite Side. 

Knowing the length of two sides is all that is needed to get the angle shown in your example. 

Now days, you don't even need to know the formula, just use an online calculator such as this; https://www.calculator.net/right-triangle-calculator.html

You know the lengths of sides (a) 6 inch and (b) 28 inch, just plug in the values and hit calculate. You don't need to know the length of all three sides all you need is any two sides.


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## woodnthings (Jan 24, 2009)

*hypotenuse length ?*

If the adjacent side is 28", the base is 6" the calculator says that "c" is only 28.6". That intuitively seems a bit short to me, but the math doesn't lie. :|


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## RichO (Apr 29, 2009)

Great. That calculator is perfect. Thanks!


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## _Ogre (Feb 1, 2013)

woodnthings said:


> but the math doesn't lie. :|


it rarely does :vs_laugh:

i'm an engineer by education. 
i believe i'm a credit or 2 short of a math degree, as are most engineers
nobody memorizes trig formulas except before a trig test
i'd google an online calculator like dave did


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## TomCT2 (May 16, 2014)

nobody does not what?


sin=opposite/hypotenuse
cos=adjacent/hypotenuse
tan=sin/cos


that and a 'scientific' calculator with inverse sin/cos/tan and no internet is required.
and before calculators, there was the CRC Handbook - which I still have....


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## Dave McCann (Jun 21, 2020)

TomCT2 said:


> nobody does not what?
> 
> 
> sin=opposite/hypotenuse
> ...


The most used item from my drafting tools is,,,,,,,,,,,,,,,,,,,,,,,,,, my eraser shield! And don't forget a slide rule for doing the calculations. 
Drafting machine? That was only for the guys with seniority, the rest of us used a tee square and triangles. I imagine there are plenty of folks who have never seen or come across either item.


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## Echo415 (Apr 3, 2018)

As promised...excuse my crappy handwriting


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## John Smith_inFL (Jul 4, 2018)

ya'll lost me way back at "then, you pick up the protractor" - - - - -

.


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## HoytC (Dec 30, 2019)

I don't think any of the calculations so far shown are correct. There's simply not enough information to calculate that angle. OP implied that the figure is a trapezoid but did not say that it is an isosceles trapezoid, which is what it appears that everyone is assuming. If that's a template made at a jobsite I don't think it's safe to assume anything about it. Just measure it.


Of course if it is isosceles then the angle is obviously the inverse tangent of 28/6 like several people have said.


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## Tool Agnostic (Aug 13, 2017)

I have never gotten over how much I use math for woodworking, especially geometry, geometric construction, and trigonometry. 

... so much so, that I keep a scientific calculator and a small precision drafting set (two compasses and a divider) in the garage. 

The slide rule lives in my desk. I don't use it any more, but I can't bear to part with it. 

I was truly blessed to have great math teachers when I was young. Somehow, someway, they made it stick. By now, they are all gone, but they did their jobs well, and they cared.


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## terryh (Nov 11, 2013)

At the risk of showing my age, here is the calculator from my university days. As you can see it has all the trig stuff anyone would ever need. I quite enjoy all the math involved in woodworking, but I don't do any with the slide rule, now it's all done with my trusty HP 11C and 15C. Can't beat RPN.


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## TimPa (Jan 27, 2010)

hoyt is correct, we are assuming that the cabinet maker is construcing a _symmetrical_ hood. but the formulaes provided are for the resulting angle, not the "compound" angle you referred in the op. 

here is a good website for the coumpound angle calculation:


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


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## Dave McCann (Jun 21, 2020)

TimPa said:


> hoyt is correct, we are assuming that the cabinet maker is construcing a _symmetrical_ hood. but the formulaes provided are for the resulting angle, not the "compound" angle you referred in the op.
> 
> here is a good website for the coumpound angle calculation:
> 
> ...


It was duly noted in post #2 that the answer given was for a simple angle NOT a compound angle.


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## RichO (Apr 29, 2009)

TimPa said:


> hoyt is correct, we are assuming that the cabinet maker is construcing a _symmetrical_ hood. but the formulaes provided are for the resulting angle, not the "compound" angle you referred in the op.
> 
> here is a good website for the coumpound angle calculation:
> http://www.pdxtex.com/canoe/compound.htm


That works great, even gives the angle to cut the miters. Too bad that page isn't a little more mobile friendly, but still pretty useful.


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## Echo415 (Apr 3, 2018)

TimPa said:


> hoyt is correct, we are assuming that the cabinet maker is construcing a _symmetrical_ hood. but the formulaes provided are for the resulting angle, not the "compound" angle you referred in the op.
> 
> here is a good website for the coumpound angle calculation:
> 
> ...


I posted the formulas for compound miters so your statement isn't correct. For those who are really lazy, they make plenty of cellphone apps that do the work for you.


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## _Ogre (Feb 1, 2013)

TomCT2 said:


> nobody does not what?
> 
> 
> sin=opposite/hypotenuse
> ...


that ↑↑↑ :grin:


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## TobyC (Apr 30, 2013)

woodnthings said:


> You can always make a layout on your bench top and just measure the angle with a digital angle finder or a protractor:
> 
> 
> 
> ...


Woodworking has been done for eons without math, especially when using wood to best advantage or fitting a given space. "Half of this", or "How many pieces can I get out of this" is common. And furniture makers in the past either cut dovetails by eye or used the chisel width to determine size. Measuring is for sissies! :wink:


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## Echo415 (Apr 3, 2018)

TobyC said:


> Woodworking has been done for eons without math, especially when using wood to best advantage or fitting a given space. "Half of this", or "How many pieces can I get out of this" is common. And furniture makers in the past either cut dovetails by eye or used the chisel width to determine size. Measuring is for sissies! :wink:


The guys framing the house across the street from me follow that logic too. Measuring isn't their strong suit. I've noticed several studs cut upto 1/2" short in load bearing walls. Sadly that seems to be pretty common these days for new construction but hey, at least nobody will consider them to be "sissies"


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## homestd (Aug 24, 2018)

Happy Fourth of July! Finding the mathematically correct solution is only half of the equation. Precision and repeatability in actually making the cut is the determiner.


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## allpurpose (Mar 24, 2016)

I'm not going to pretend to know squat about mathematics, but at least I didn't look at a business card that has "carpentry" printed on it then ask, "Oh! So you do carpets?" 
That actually happened to my son yesterday.. lol

"Carpet builder!"


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## woodnthings (Jan 24, 2009)

*They make compound wood hoods .....*



RichO said:


> *At the cabinet shop I work at, we build compound mitered wood hoods for many homes.* While there are different ways to build one, the guy that heads my department draws it out full scale on a sheet of wood and then uses a digital protractor to get the correct angle to cut the pieces at.
> 
> I am not much of a geometry guy but there must be some kind of formula you can use to get the angle based on the measurements, like in the attached drawing.
> 
> ...





TimPa said:


> hoyt is correct, we are assuming that the cabinet maker is construcing a _symmetrical_ hood. but the formulaes provided are for the resulting angle, not the "compound" angle you referred in the op.
> 
> here is a good website for the *compound angle calculation*:
> 
> ...





Dave McCann said:


> *It was duly noted in post #2 that the answer given was for a simple angle NOT a compound angle.*



So, which is it, compound or straight/vertical sided hood? The angle won't change in the view that was first posted, BUT If the sides are not vertical it will lean away from vertical and the angle will change slightly, at least that's what my intuition tells me. :|
Probably the best way to do this is via Sketch Up, which I don't know how to use ...... where you enter the height, length and width values and it will calculate all the angles.


The title was misleading:
Calculate angle from drawing

"Calculate" implies using math. 

"from the drawing" could mean either using math or a constructed 

diagram with just measurements.
If the angles are compound, then I would use the online calculator.
:vs_cool:


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## phaelax (Dec 24, 2018)

Assuming this is an isosceles trapezoid (symmetrical), using the height to find the angle in the triangle at the bottom right, 77.91 degrees. Subtract from 90 to find the angle at the top of the triangle. As the angle adjacent to this is a right angle, add 90 to get the obtuse angle of the top of the trapezoid.


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## RichO (Apr 29, 2009)

I do have a follow up question regarding one online angle calculator someone posted a link to.

I just built a compound hood this week using this page to come up with the angle. However, I am puzzled by what they are referring to with the "End Angle". I was expecting that to be 90 minus the slope angle (21.801) which is the angle I used to bevel the top and bottom of the "box", but what angle or cut is the 20.374 supposed to apply to? Any ideas? 

Thanks


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## woodnthings (Jan 24, 2009)

*The end angle can be ......*

The end can be at angle you choose from straight up vertical to the same as the sides depending on your hood design. Most of the online calulators I've (not used) have the same angle on the ends as on the sides making the compound cuts identical. It would certainly throw a slight "curve" (not really) in the equation if the end angle is different. do ask me what to do in that case. I have made similar objects including a roof for a wishing well with 8 sides just by a "cut and fit" method after establishing the height. I then set the fence on the jointer to make the joint more precise. FYI, I enjoy math but was never much good at it. I flunked analytic geometry in college and that was a 5 credit course ..... :sad2:


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## TimPa (Jan 27, 2010)

i am thinking it is the angle that the top edge (end grain) and bottom edge (end grain) would be cut at to be horizontal when assembled


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## RichO (Apr 29, 2009)

TimPa said:


> i am thinking it is the angle that the top edge (end grain) and bottom edge (end grain) would be cut at to be horizontal when assembled


That was my original though but that angle should be 21.8 degrees, always 90 minus the slope of the hypotenuse.

woodnthings, I get what you are saying, and personally I prefer to slightly adjust the angles of the horizontal ends as to get a nice tight fit at the top and bottom joints. It's just that this calculator seems to be based on geometric calculations to achieve a specific end result so that is where the 20.374 is confusing.


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## woodnthings (Jan 24, 2009)

*Correction/edit ....*



woodnthings said:


> The end can be at angle you choose from straight up vertical to the same as the sides depending on your hood design. Most of the online calculators I've (not used) have the same angle on the ends as on the sides making the compound cuts identical. It would certainly throw a slight "curve" (not really) in the equation if the end angle is different. DO NOT me what to do in that case. I have made similar objects including a roof for a wishing well with 8 sides just by a "cut and fit" method after establishing the height. I then set the fence on the jointer to make the joint more precise. FYI, I enjoy math but was never much good at it. I flunked analytic geometry in college and that was a 5 credit course ..... :sad2:



Corrected wrong info.....


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## TimPa (Jan 27, 2010)

RichO said:


> That was my original though but that angle should be 21.8 degrees, always 90 minus the slope of the hypotenuse.


if you go back to the web page and read more, you will see what i am talking about.


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