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Carpentry and Mathematics

12/30/2014 9:16 AM

Does anybody remember a discussion some years ago? It was about two down-sloping "shelves" meeting at an external corner, and the question was how to calculate the angle to saw across the shelf, and a more difficult one, the angle through the thickness, so they would meet correctly. I can't remember how far we got last time.

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#1

Re: Carpentry and mathematics

12/30/2014 9:33 AM

That called a compound angle formula. (just the math aspect)

Usually happens/needed with trim or making picture frames.

I did my bathroom trim, takes some experience... of which I didn't have, But I did have enough experience to not make my inexperience look noticeable. Owe that to a very good miter saw and practice cuts.

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#3
In reply to #1

Re: Carpentry and mathematics

12/30/2014 11:41 AM

Thanks but I'm not too sure the compound angle formulas cover it.

I don't want to do the carpentry, but one way would be to calculate the cut angle across the board (simple bit of trig), mark the line and then support the board at the correct tilt angle. Then do a vertical cut through the line, OK as the ends of the boards clearly meet in a vertical plane. Saves the more difficult calc.

I first came across the problem about 25 years ago for a 90° corner, and any slope angle. Somebody had asked the local maths teachers and none could work it out! Using vectors, I came up with a formula, for the general case of any corner angle, which I'm 95% sure is correct, but wanted to compare with the discussion on CR4.

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#4
In reply to #3

Re: Carpentry and mathematics

12/30/2014 12:17 PM

That link... its not for a carpenter...

I was fortunate to have a good supply of wood to do some practicing. And how I did it was also using vectors also. Some way I used to work out static problems in College. I had to do it one angle at a time, then move to the next angle.

Unfortunately, when I sawed it on the cut off saw, I set the blade tilt wrong. I forgot...measure twice, cut once.

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#34
In reply to #3

Re: Carpentry and mathematics

02/03/2025 1:07 PM

this is old post, I did a project (cupula) where I wrote an excel file that calculated the compound angles with a given dimension.

You can download the excel file from here.

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#35
In reply to #34

Re: Carpentry and mathematics

02/04/2025 9:51 AM

By a remarkable coincidence, I'd just re-opened this old thread myself!

You've done an impressive piece of work there, not just the building itself but the detailed write-up.

I revisited it just out of interest, to confirm I had the formulas right. Pleased to say putting the figures from your excel into mine gives same answers.

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#36
In reply to #35

Re: Carpentry and mathematics

02/04/2025 2:59 PM

i came across this old post too.

As far as the angle, It’s funny you did that, I was always going to check the angle to make sure it the angle of the blade from 90 degrees. I checked it a number of times it was getting late so and Instead I sawed it, and it fit, lucky guess lol.

ie,… the blade to be set at 48 degrees as the program states the opposite angle of 43 degrees.

one thing about that, I had bought a Incra miter for my table saw, it’s accurate to 0.1 degrees, which was good enough for me, lol

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#2

Re: Carpentry and mathematics

12/30/2014 10:30 AM

I think it easier to make a pair of jigs to support the boards so that the saw blade is only moved in one angle. The jigs are made with the tilt angle incorporated into the supports. For a finished frame that looks square face-on, the angles B would be +/- 45 degrees.

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#11
In reply to #2

Re: Carpentry and mathematics

12/30/2014 11:43 PM

Wait...

Did he say, "...jigs..."?

I think I'm offended.

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#5

Re: Carpentry and mathematics

12/30/2014 3:18 PM

The face angle is defined here as the angle C. This is found as shown.

When B is 45: and

A is 0, C is 45 degrees;

A is 30, C is 40.893 deg;

A is 45, C is 35.264 deg;

A is 60, C is 26.565 deg;

A is 90, C is 0 deg.

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#8
In reply to #5

Re: Carpentry and mathematics

12/30/2014 6:05 PM

I think I understand your sketch and that A is angle of shelf below horizontal.

I can't work out what your B is. It's obviously 1/2 the 90° in the case of a 90° bend, but what is it for other angles? In my formula below B is 1/2 the corner angle, defined like a pipe bend, deviation from straight, so a 45° has 135° included angle.

C is the angle of the cut across the board, from the normal (i.e. for a cut straight across, C = 0). My formula is C = atan(cos(A)*tan(B)), which gives the same results for B = 45°, but not for other values using any way of defining B that I can think of.

Also you don't mention the included angle between the board ends, hence the angle of cut through the board thickness (= 1/2 included angle). That's the more difficult bit!

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#9
In reply to #8

Re: Carpentry and mathematics

12/30/2014 8:57 PM

B is simply the miter angle. If you image, say, crown molding wrapping around an exterior 90 degree corner it is the half-angle of that corner. Or if its a rectangular picture frame it's half the 90 degree interior angle.

Crown molding is cut at a 45 degree angle, but the wood being cut must be held in the miter saw at it's 'installed' angle, which yields a compound cut.

If I understand what you're asking about the angle of cut through the board thickness, it would be 135 degrees when measured perpendicular to the line of cut (or 45 for an exterior angle) . The two faces form a 90 degree angle when the facing boards are placed together. (Image the corner of a shoebox; it doesn't matter what angle the shoebox is held at, the two sides form a 90 degree angle). The two boards can be imagined as meeting at the corner of the box.

The key is what I called the face angle C in my diagram. It is given by the formula in my diagram. (But as I said in my first post, it's easier to make a jig to support the boards at the finished angle (or, mount them in the miter saw at the finished angle).

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#18
In reply to #9

Re: Carpentry and mathematics

12/31/2014 12:43 PM

OK I understand B. It relates to what I called the bend angle X by X = 2*(90 - B). Putting that into my formula for C becomes C = atan(cos(A)*tan(90 - B)). But I can't agree with your formula for C. When B = 45, we agree for any value of A, but for any other B, we're different, and yours gives imaginary answers (in Mathcad) or #NUM! in XL for some quite reasonable inputs.

By the angle of cut Z through the board thickness, I mean the angle to cut so the ends of the boards meet without a gap, = 1/2 the included angle between the board ends before cutting. For a 90° corner and A = 0°, my formulas give Z = 0° (i.e. straight through) and C = 45°. For A = 90°, Z = 45°, C = 0° (i.e. straight across). All as expected.

The two faces only form a 90 degree angle, with line of cut 45°, when A = 90°. Different at other values of A. Perhaps we're getting at cross-purposes!

Mitre boxes are OK for carpentry, but if we have say two fabricated steel frameworks at any slope which need to meet at a corner of any angle, with pads bolted together, it would be good to calculate the angles.

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#20
In reply to #5

Re: Carpentry and mathematics

12/31/2014 7:10 PM

I used to do sloping fascias on these;

You can just see the white timber under the sheeting. In your diagram the 60 deg slope would equal a 30 deg pitch as is this roof. I remember that i also had to calculate for a side cut. So i would set my saw to the cut angle; as per your calculations 26.56 deg ( sounds wrong but i will work it out soon ) and then tilt the saw blade over. In my memory about 12 deg.

To do this i used trig. It was made harder by having a 6 or 8 sided structure.

Jim

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#22
In reply to #5

Re: Carpentry and mathematics

01/01/2015 5:24 PM

Had another look at this and the closest I can get to your formula

cos(C) = [tan(B) / cos(A)] / sqrt [1 + tan(B)2 + tan(A)2]

is cos(C) = sin(B) / cos(A) / sqrt [1 + tan(A)2 * sin(B)2]

which gives same results as my tan(C) = cos(A) / tan(B) using your definition of B.

My formula found as follows.

For shelf of unit width, projected width in plan = cos(A). So "extra" length of board along outside edge (easier with a sketch but I haven't got the hang) = cos(A) / tan(B)

This is actual length to saw to on the full board width, so tan(C) = cos(A) / tan(B).

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#6

Re: Carpentry and mathematics

12/30/2014 3:20 PM

With a compound miter saw it would be easy. Swing the head to half the external angle and tilt the head to what ever slope you want on the shelves. You just have to reverse this for the cut on the other board.

Use some junk wood to practice on. Sometimes what the measured angle is at the corner. Is not what the wall is.

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#7
In reply to #6

Re: Carpentry and mathematics

12/30/2014 4:03 PM

Won't work.

If you set the miter angle to 45 and keep the blade at 0 tilt, you'll get a normal square corner, but if you set the miter to 45 then tilt the blade, the net result will be face angle of less than 90 degrees for the assembled frame.

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#10
In reply to #7

Re: Carpentry and mathematics

12/30/2014 11:08 PM

The tilt angle will also be less than what was set on the saw.

There are plenty of compound mitre calculators on the web, some are not so accurate so put your numbers into a few of them and go with the consensus.

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#15
In reply to #7

Re: Carpentry and mathematics

12/31/2014 8:54 AM

As I have done this before and just went and did it again. I set the miter angle at 45 degrees and the compound angle to 10 degrees. I cut the wood to form a exterior 90 with a down sloping angle. Comes out when assembled to good 90 to my square. Nothings perfect. Saw could use a sharp blade and the blade angle gauge has never been set. And I could have clamped the wood to make the cuts. Wood was just plywood strips. 1" X 5.5"

Saw was purchase to cut up scrap.

Also used CAD made a boards and the blade 3D. Set the angle of the blade as above thru my boards. Subtracted them and put the boards together. Now that was perfect.

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#19
In reply to #15

Re: Carpentry and mathematics

12/31/2014 1:45 PM

Your method may be OK for rough work, but it's not accurate.

Setting the mitre cut at 45° and the compound at 10° will result in an included angle of 88° not 90°. May not sound like much, but that's enough to put the shelf under square by around 25mm in 600mm (1" in 24" imperial), that's a 4% error and easily enough to wreck a job that may require accuracy. Also, the included slope angle will not be the 20° that you might expect from the 2 x 10° cuts but rather closer to 14°. Consider making a 4 sided box with this amount of error, the last bit would not fit without opening the joints.

If going for better accuracy to achieve 90° and 10° slope, you would set the mitre to 44.5° and the compound (tilt the blade) to 7°.

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#17
In reply to #6

Re: Carpentry and mathematics

12/31/2014 11:49 AM

The angles for compound cuts for crown moulding, on a miter saw are given on the saw, and are generally set as "stops".. THere are two angles that are most used for crown moulding, one surface is against the wall and one is against the ceiling. The angles of the stops correlate to those. They can be flipped for aesthetic reasons. As for the compound angles necessary for the sloping shelves, use the "jigs " mentioned, and set them up on a table saw and run the shelves through that way.One thing to look out for, is that one uses a sloping angle , the width of the boards generally narrow as you go down, which changes the problem. If not, then good to go… As previously mentioned, one can mess around with formulae all day, but using mock ups are generally what gets the job done, in the long run….

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#32
In reply to #6

Re: Carpentry and mathematics

01/07/2015 5:33 AM

You just have to reverse this for the cut on the other board....

OH YEAH? Not so easy!! It may be less confusing to do the maths...

So if I continue to hold the saw in my right hand and turn to face the opposite direction, this must be the reverse...hold on, didn't I turn that plank over just now ? Surely if I did turn aropund, the saw should be in my left hand ? !@$#%^&$CR4!! where's my calculator now?

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#12

Re: Carpentry and Mathematics

12/31/2014 5:12 AM

Deliberately putting up wonky shelves sounds like an excellent ruse for getting your wife never to ask you to put up shelves again. Making the corner neat so that the wonkyness is less noticeable rather defeats the object of the exercise. Cutting the angles at a random angle that is near enough makes it a cinch that she will employ a competent carpenter next time, leaving you in peace.

Happy new year everyone.

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#13

Re: Carpentry and Mathematics

12/31/2014 6:04 AM

A compound formula can be developed because I did it about 40 tears ago, long before calculators and computers, for a woodworking friend who wanted to make cone shaped hoppers and bins and frames of various depths.

I can't find the paperwork now (two house moves since then) but Usbport 9 trig formula looks something like it. Although in my case I had to take account of the number of sides required as well as the angle/slope of the sides, to give the length and angles to cut each piece, together with the angles to produce a 'flat' face on the edges of the back and front pieces (or top and bottom) when assembled.

The formula worked a treat when the sides required were a whole number, but the fun was the shape produced when you input decimal value sides in the formula - like 4.5 sides.

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#14
In reply to #13

Re: Carpentry and Mathematics

12/31/2014 8:08 AM

Interesting. Sounds like fractals.

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#16

Re: Carpentry and Mathematics

12/31/2014 9:11 AM

A 'carpenter' would use a framing square, level and a pencil to calculate this compound cut. It's amazing what you can figure out with the information on a framing square.

I was a carpenter before I went to college. When I took Trig in college, I could figure out problems very closely with my framing square. It took a long time to change my thinking to Trig formulas.

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#21

Re: Carpentry and Mathematics

01/01/2015 11:57 AM

1. Measure twice

2. Cut once

3. Buy more wood

4. Repeat until correct.

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#23

Re: Carpentry and Mathematics

01/01/2015 7:02 PM

I can't claim to be a mathematician and definitely not a carpenter, but you definitely need a miter box and some way to clamp the shelf pieces at the correct angle. Calculating mathematically the correct angle and then trying to cut at that angle is too difficult, IMHO. If the shelf pieces are too big for your miter box, possibly you could cut smaller pieces and use them as a guide for cutting the shelf pieces at the correct angle.

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#24

Re: Carpentry and Mathematics

01/01/2015 8:36 PM

As a PS to my previous post 13

I can say my friend tried my formula but could not cope with the maths. He gave me the number of sides and slope and I calculated the angles for him (slide-rule and trig table sin those days) and he set the jigs. He practised on tiny pieces of wood to confirm the angles were correct.

With modern cross-cut mitre saws it would be dead easy without complex formulas. You swing the cross-cut angle to give you the slope you want, and you tilt the saw to the angle that makes the ends butt correctly.

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#25
In reply to #24

Re: Carpentry and Mathematics

01/01/2015 9:11 PM

You guys are making this all too complicated. As I said earlier, there are plenty of charts that do all of the calculations for you available on the web.

Here is one that I consider to be the best.

Look here

It covers just about any combination of angles that you may desire and calculates them all for you.

For the two intersecting shelves that are the subject of this discussion, go to the site and then down to the section "General compound angles 2"

Enter the horizontal angle, in this case 90°, and then the slope angle of the shelves in the other 2 boxes. In the example below I have used 10°

The calculator will give you a mitre angle of 44.6° and blade tilt to 7°, set the saw and cut the board.

In this case you don't even have to alter the saw to achieve the entire joint, just allow enough material each side of the single cut for both intersecting shelves, turn one over and they will mate perfectly for your 90° angle and 10° slope.

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#26
In reply to #25

Re: Carpentry and Mathematics

01/02/2015 4:41 AM

Thanks for this link. A hearty congrats to the man who made it available. The box of N sides is the same as the sloping fascias in my post #20. I made an excel spreadsheet with trig formulas so I could do the same thing. In those days the formulas were based on radians so the degree field had to convert to radians and then put the answer in a new field. The formula then had to refer to the 'new' field. Took a while but it saved time and made for a better job.

Jim

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#27
In reply to #25

Re: Carpentry and Mathematics

01/02/2015 8:05 AM

For a bit of fun, go to your 'look-in' link and use the 'N-sided Box' formula to calculate the angles of a box with say 3.5 sides (or 4.5 sides, or any fraction come to that).

Cut the wood, then assemble to see what the finished box looks like. The result might not be what you expect.

I do not have the tools here to do it myself, but experts with CAD facilities , with time to spare, might be able to use the angles to draw the finished box - and post the drawing here.

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#28
In reply to #27

Re: Carpentry and Mathematics

01/02/2015 9:10 AM

Yes, I noticed that. I wondered if it covered the general case of any corner angle, since there's no reason to restrict it to corners of an n-sided polygon.

For any corner angle X (defined as deviation from a straight line), if you enter Number of sides = 360/X to as many decimal places as you like, it gives the right answers (or at least my formulas give same results, as they do for other inputs).

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#30
In reply to #28

Re: Carpentry and Mathematics

01/02/2015 11:14 AM

As the formula can handle 'fractional' numbers of sides, I am curious as to what final shape you get.

For instance, what does a box with 3.5 sides look like ?? - when assembled to the angles the formula gives for 3.5 sides??

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#31
In reply to #30

Re: Carpentry and Mathematics

01/02/2015 1:12 PM

It doesn't have to have a fractional number of sides. It's just that we might need to fit the shelves to a corner of any angle, it needn't be a corner of a regular polygon.

That's why I prefer to designate it by the bend angle X from straight, like a pipe bend. Using the link, need to "trick" it by putting Number of sides = 360/X. Of course, doing it my way, if it is a regular n-gon, need to work out X = 360/n, but that's not a big job.

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#29
In reply to #25

Re: Carpentry and Mathematics

01/02/2015 10:24 AM

Agree with others it's an excellent link. It confirms my formula for the angle Δ to cut the board ends, which for anybody interested is given by

cos(2*Δ) = sin(A)2*cos(D) + cos(A)2 where D is the angle of the corner, deviation from straight.

Thanks everybody for the help on this one, and Happy New Year!

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#33

Re: Carpentry and Mathematics

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