why is a 6/12 - not a 22.5 degree angle ?

[speaker]

Hello.

I was wondering if someone could tell me why a

12/12 pitch roof is a 45 o angle

but a 6/12 pitch roof is 26.57 o angle?

thank (agaim)

[Tom R]

You ask this question first thing on a Sunday morning? - - I like your style. I'll vounteer the first 'honest' answer, - - I don't know, - - but I've wondered it myself. I'm sure it's got to do with the trig-a-ma-jig thing, - - sines, co-sines, tangents, co-tangents, and other words in the math-bath, - - can't wait to hear if someone can give a good answer.

[speaker]

(lol) Ya trig a ma jig no doubt, I would just like to know why I am doin' things (or at least have an idea, of why I'm doin' it)

[Tom R]

Yeah, that's a good 'sine'. :D

My money's on Joe Carola for this answer. Joe- - - HELLLPPPP!!

[Tom R]

The closest I get to trig is when I 'co-sine' a loan! :D

[Rich]

It really comes down to some geometry and trig. When looking at the complete arc - the bisector works out to 5/12 but when just looking at a triangle with sides of 12 and 12 - half the vertical distance of a 12/12 is a 6/12 - but you're not looking at the full picture. Think arclength and bisectors. I won't attempt to explain the trig involved - would get too confusing :)

[speaker]

O.K. rich fair enough
but, is there a theory without getting into the trig, just so as I might have somekind of understanding?

[Rich]

Like I said - it comes down to an angular bisector instead of a dimensional bisector. Look at the two conditions in the attached pdf.

[Rich]

This pdf is probably easier to visualize.
Notice the 6/12 is not the chord bisector - it's the triangular bisector as shown in the last pdf. The 5/12 is the true chord bisector.

[Tom R]

I'm on my way out the door, - - but I'll 'venture' a simplified explanation. Although it 'sounds' like 6/12 is exactly one half of 12/12, - - it's actually not because the '6' leg is not at the correct angle to bisect the arc that is formed. The 'correct' angled line would be at a right angle to the newly formed hypotenuse (22 1/2 degree line), - - or when you 'halve' a line connecting the hypotenuse with the long leg of the right triangle (with equal acute angles) at any point. It's 'this' line that would be cut in half to form half of any given angle. (Don't know trig, - - but I have a good understanding of Geometry, - - the foundation)

Can anyone understand this jibberish? - - I got it pictured in my head, - - but it's hard to translate it on paper.

See, no wonder my wife tells me I'm 'simple' :D

[speaker]

Thanks guys for your follow ups (my persistance paid off) I think I understand it. (not the calcs., but the reasoning)

Rich - on your 1st pdf the vertical bisectors look to be the same length (hense the rise 6/12 and 12/12 (double the rise) but on the angle they are not the same length. because of the arc, thank you

Tom, and your explaiation in writting helps clear it up also
both of you, again thanks alot!

[Joe Carola]

Speaker,

You have been given great answers so far and a lot better then what I could have given you. How do you figure your rafters . Do you use a Framing Square, Calculator, Scientific Calculator, Construction Master?

Joe Carola

[speaker]

You are correct, I have learned a ton here.

I framed houses in the Toronto area for 6 years, (15 years ago, when we had to hand nail everything) but when we did roofs there were mainly trusses, and even when we did hand cut roofs I was a rafter assembler not calculator.
Now, I am building a cottage, and would rather not wing it.

So to answer your question, I will know the pitch of my roof and run, so I will use a scientific calculator to figure plumb line, hap, commons hips etc... and double check on the square.

with out this site though, I would not been ready to go!
and I still want to learn more...

what about yourself?

[Tom R]

Here, everybody, - - I did a little drawing in hopes to better explain the voices in my head. :D Hope this helps. Shown are the two different 'mid-point' locations of 6/12 versus 22 1/2 degrees. Note that the bi-secting line is at a complete different angle than the 'rise' line. This is where the discrepancy lies in the 'half' theory (half of the angle vs. half of the rise).

[Rich]

Not trying to diminish anything you've done Tom - but isn't that pretty much the same thing I posted :)

[Tom R]

Yeah, Rich, - - never hurts to have another 'perspective'.

Besides, - - it's like personal therapy, - - It's really ME that needed another picture, - - I sometimes pose as 'everybody'! :D

[Rich]

LOL - ok - just checking.

BTW - How many of you are there in your personal therapy :)

[Tom R]

We've never quite agreed on the real number. Just know it causes a fight everytime. :D

[Tom R]

Hard to know if you've won or lost, - - when your hands and face have the same amount of bruises! :D

[Tom R]

Most of us appear to be right-handed, - - but apparently us left-handers can pack a 'wallop' too! :twisted: OUCH!

[Tom R]

Say Goodnight, All! :wink:

[speaker]

Yes as Rich says it is the same, but with a twist. Because the pitch and the angle are situated within the circle, you can see the big picture.

I do have 1 question though, how did you come up with the angular line that shows the "midpoint of the 22.5 degree angle? in other words, why is the line on that particular angle?

thanks alot, very helpful!

[Tom R]

That line is formed by a drawing a perfect right-angled 'T' with equal top lengths, coming off of the angle where the 'run' and the 'hypotenuse' meet. You can see the 'T' in the picture, the bottom of it starts in the bottom left of the picture (the long leg of the 'T' is the bottom dashed line, and the top of it is the bisecting chord). The 'T' can be any length, as long the top of it has equal lengths, it will be 'centered', and therefore cut the angle in half.

[Tom R]

You can actually make the 'T' by setting your compass to any length, putting the point at that run/hypotenuse intersection, and then drawing an arc that connects the run to the hypotenuse at any point. Then you just draw a straight line to connect the two points that the arc intersected (this line will actually be the top of the 'T'). This method will automatically make the top of the 'T' at the correct angle. Then the 'centering' of this line (the bisecting chord) will be your 'half-angle' mark.

[Joe Carola]

Why are you using a calculator to figure your H.A.P. cuts and Plumbcuts.

I just mark the plumbcut out with the framing square and then hook my tape and mark the length of the rafter and make the birdsmouth with a seatcut of 3-1/2" for a 2x4 wall and whatever the H.A.P. cut is it is. I've never used the 2/3 rule for my H.A.P. cuts before or ever heard of it before until I saw it in a book and it's been mentioned on the internet. Same goes for a 2x6. I've never had a problem.

This is something I posted to someone about using a Construction Master Trig for figuring a Plumbcut, Birdmouth and H.A.P. cut.


A birdsmouth is a little Triangle for example a 5/12 pitch using 2x8.

5 [Inch] [Pitch]
3.5 [Inch] [Run] (2x4 wall)
Press [Diag] Returns - 3.791667" or 3-13/13"
Press [Rise] Returns - 1.458333" or 1-7/16"

Plumbcut for a 5/12 using .

5 [Inch] [Pitch]
7.5 [Inch] [Run]
Press [Diag] Returns - 8.125" or 8-1/8"

H.A.P./HEEL cut.

8-1/8 - 1-7/16" = 6-11/16"

Or do it this way using Trig with the same calculator.

Birdsmouth.

5/12 = [Conv] [Tan] Returns - 22.62°
3.5/22.62 [Cos] = 3.79167" or 3-13/16"
22.62 [Tan] x 3.5 = 1.458343" or 1-7/16"

Plumbcut.

5/12 = [Conv [Tan] = 22.62°
7.5/22.62 [Cos] = 8.125" or 8-1/8"

H.A.P/HEEL cut.

8.125 - 1.458 = 6.667" or 6-11/16"

With those numbers using the run to the outside of the plate and deducting 1/2 the thickness of the ridge you can just add the H.A.P/HEEL cut to the rise and that's your ridge height.

If you use the run to the inside of the plate as I do you deducting 1/2 the ridge thickness just add the plumbcut measurement to the rise and that's your ridge height. Doing it this way is less steps and faster (as I know you would want Blue) you don't have to figure the birdsmouth first.

The point of doing all this is that you can do it at home on a piece of paper before you get to the job or just lay it out with the framing square or if someone here asks the question which people have you can just do it without using a framing square in front of the computer.

Speaker, I'ts good to know how to use a calculator of your choice especially when it comes to cutting Irregular Hips but are you just using the calculator to figure the plumbcuts and H.A.P. cuts for a common rafter exercise only?

Joe Carola

[speaker]

Tom, Thank you, It's quite obvious (in hind sight, - that is) appreciate your patience!

Joe, To have the top plate completely covered by the seat, you really don't need to calculate anything at all, right? just use the square on the plumb line, untill the plate thickness remains on the tounge.

but I ran accross this site (I poted it before)

http://www.josephfusco.org/Articles/Roof_Cutting/raftercutting.htm

This guy calculates everything, I was interested in the calcualtions, though I discovered that his calcs. for plumb line and HAP can be calculated using cos instead of sin which removes one step from the process, which is (90- your roof angle), all you have to do is punch your roof angle in (new to me)

below is the calcs. you sent to me, I am interested in these, but what is (diag) in your first example)
(conv) in your second example

thanks for all the work you did!

A birdsmouth is a little Triangle for example a 5/12 pitch using 2x8.

5 [Inch] [Pitch]
3.5 [Inch] [Run] (2x4 wall)
Press [Diag] Returns - 3.791667" or 3-13/13"
Press [Rise] Returns - 1.458333" or 1-7/16"

Plumbcut for a 5/12 using .

5 [Inch] [Pitch]
7.5 [Inch] [Run]
Press [Diag] Returns - 8.125" or 8-1/8"

H.A.P./HEEL cut.

8-1/8 - 1-7/16" = 6-11/16"

Or do it this way using Trig with the same calculator.

Birdsmouth.

5/12 = [Conv] [Tan] Returns - 22.62°
3.5/22.62 [Cos] = 3.79167" or 3-13/16"
22.62 [Tan] x 3.5 = 1.458343" or 1-7/16"

Plumbcut.

5/12 = [Conv [Tan] = 22.62°
7.5/22.62 [Cos] = 8.125" or 8-1/8"

H.A.P/HEEL cut.

8.125 - 1.458 = 6.667" or 6-11/16"[/b]

[Rich]

Joe was talking about using the Construction Master calculator - [Diag] is a key on the calc.
[Conv] is like second function on other calculators.
And you are right about not needing the calculations figuring hap - framing square does just fine.

[Joe Bartok]

Well, a little late, but I'll try to answer the half angle question.
Reference any of the posted diagrams (saves me making one up for this purpose). There are actually two ratios involved: one is the tangent of an angle, or the rise/run. And, although most people don't think of angles this way, the other ratio is the angle itself. The true definition of an angle is the circular arc length divided by the radius of the circle. The resulting ratio is the measure of the angle in radians, and any calculator or computer program such as a worksheet or Javascript actually uses radians to calculate angular related quantities. The radian values are generally converted before being displayed by multiplying by 180/pi; this results in values in degrees that most people are more familiar with.
Taking this into consideration, have a look at the diagrams. If you imagine halving the arc length shown (in other words, halving the angle), you can see that the rise/run can't possibly also be halved. Try making a sketch for yourself, using a large value for the angle, say 90 degrees.
If any of this isn't clear, just say so, and I'll post a sketch tomorrow.
Speaker- I revisted the josephfusco link, and it seems he calculates the HAP both ways discussed in a previous thread. And, as mentioned above, calculators and computers are great tools, but not always necessary. Carpenters and timber framers in the past had nothing to rely on but their knowledge of geometry, ratio and proportion, and many of their creations are still standing to this day, hundreds of years later.

[speaker]

Joe I am yet to scrutinize your reply,

but, rich what I meant was, once you have calculated the hypoenuse (ridge peek to plumb line over outside wall), draw the plumb line in, then slide the square with the body mirroring the plumb line until the tongue measures 3 1/2" (assuming a 2x4 plate), and then draw your seat.

wouldn't this work?

[speaker]

I thought that the "DIAG" might have been on some type of construction calc., but the "CONV" you said
[Conv] is like second function on other calculators
what does that mean?
I was using the "window" scientific calculator, it has "sin" "cos" etc.

(you may need patience for this one)

[Tom R]

I beleive the DIAG is for diagonal, or hypotenuse

And CONV for convert, or conversion

These are on a construction calculator

[Joe Bartok]

Well, I can't get to the pdf creator I downloaded the other day, so I'll try attaching a .doc file. Also seems I'm also out of sync with the current topic (sorry!).
This diagram is pretty much the same as what's already been posted, but with a bit of a different perspective. Now, if the angle were bisected (actually, it doesn't have to be a bisect, use any proportion you wish), is the rise/run as shown even close to being bisected. No way!
Hopefully this answers the original question using just geometry.
By the way, because of the fact that the angle and its related rise/run don't vary equally, be careful when calculating the tangent or rise/run for the sum or difference of two angles. You need a formula. If requested, I'll pass it on tomorrow.

[Tom R]

I couldn't get it to download, Joe, - - but them I'm not too computer-savvy. Maybe others still can. I would like to see it.

In the meantime, here's another diagram I did to show how to arrive at the 'half-angle' point.

Set compass at Point 'P'
Draw arc 'A' and arc 'B'
Set compass at Point 'A' and draw arc 'AA'
Set compass at Point 'B' and draw arc 'BB'
Connect the Points

Any compass setting will work as long as you do both sides with the same setting.

This method will automatically 'split' your angle perfectly.

[Rich]

Why do the bisector that way if you already have the chord there (Line AB). Just half the length between A and B. Also if you move the point p origin to the center of the circle you shortcut another step by not having to find point A (wherever the 45 line crosses circle with radius PB).
Anyway - just adding some shortcuts there.

[Tom R]

You're right, Rich, there's many different ways to do it, and your short cuts are well-noted, and of course, correct. I had already mentioned in an earlier post just measuring to the middle, this is just a different method (hopefully mind-stimulating), but also Speaker had asked in an earlier post how to determine the exact 'angle' of the chord. Just trying to kill a couple of birds with one stone.

P.S. This method was to 'establish' line AB before it was there, and then to establish the 'half-angle' line. I'm drawing the points first, then using them to establish the lines. In other words, essentially starting from scratch (just the 12/12 triangle). In actuality, even the circle isn't necessary, - - I just feel it helps people 'see' the big picture.

P.S.S. Man, you're tough :shock:

[Joe Bartok]

Tom: I tried downloading my doc file, and it works OK at my end. Sorry I can't do it today, but, tomorrow, I'll make a pdf file and repost. Actually, my diagram is very similar to the one you just posted. Take your drawing, and draw an arc, from the upper vertex of the triangle to the hoizontal line, with the center of the arc at the left hand vertex of your triangle. Now bisect this arc (which is actually an angle); you'll see that the rise of the triangle is nowhere near close to being bisected.
Actually, you don't have to even make another drawing, in essence, what I've just described is what's happening in the drawing you've posted. Geometrically, you can see that if you bisect your angle, there's no way the rise can be bisected.
Wow! I should have stayed on-line all day just to keep up with you guys!
This keeps up, Rich might have to make a separate section just for math questions. :)

[Tom R]

Yeah, Joe, though I'm definitely no expert at it, I've always been somewhat 'addicted' to math, especially Geometry. And you know what, - - a 'Math' section really sounds like a great idea!

Whaddaya think, Rich? - - wanna 'add' it in? :wink: :D

[Rich]

I think that would be a great addition. And not meaning to be tough on you Tom - just trying to get as many different looks and explanations as possible for everyone. I'll get that section added.

[Tom R]

Yeah, that's fine, Rich, - - it's always healthy to 'spar' the choices. In fact, check Team Contractor, I'm 'sparring' over there with you right now. :D
Seriously, though, yeah, - - a math section would be great!

[Rich]

Done - in the Engineering / Arch section.

[speaker]

Well, I just got home and wow.....I would just like to say I got it. makes sense now. and maybe we'll see ya in math class I guess?

Thanks to everyone (Iknow that you were having fun, but thanks just the same)

[Tom R]

Glad we could all help, - - yeah, it was fun. See ya in the next class :D

[giddonah]

Math proofs are usually done with few shortcuts on purpose. In the first semester of calculus you learn how to do it the hard way. The next two you learn how to do it without anything from the 1st semester. It seems like a wasted semester, but it's the foundation that gives understanding. Learning just the shortcuts limits the usefullness of the information. I liked everyone's drawings, but it was Tom's that finally set it for me. Seeing his first might not have done it though. The best text books explain everything in at least three different ways so at least one of them sinks in.

Just a little rambling. Thanks for the info.

[Joe Carola]

I thought that the "DIAG" might have been on some type of construction calc., but the "CONV" you said
[Conv] is like second function on other calculators
what does that mean?
I was using the "window" scientific calculator, it has "sin" "cos" etc.

(you may need patience for this one)


Speaker,

On your windows calculator all you have to do is.

5/12 = 0.41666666666666666666666666666667

Then click in the square at the top left that says Inv and then click on tan and it will return. 22.61986494804042617294901087668

So round that off to 22.62° for a 5/12 pitch roof.

On my Construction Master Calculator I have to press the [Conv] button to achieve that.

The [Diag] button on my calculaltor is the same thing as Hypotenuse.

As far as the Birdsmouth yes just mark your length at the outside plate line and mark a plumb line with the framing square and then slide it down until it reads 3-1/2" level for a 2x4 seatcut that's all.

Joe Carola

[speaker]

Joe that is one thing I really wanted to know, It just gives me a better understanding of the reasoning. and as far as the birds mouth, That's great, just cut'n'go
thanks Joe

[Joe Bartok]

Wow! I haven’t had a chance to really test drive this pdf creator, but I like it already!
Here’s the diagram I posted yesterday, this time in pdf format.
I had a chance to consider speaker’s question overnight, and thought of a bit more related information to add. Included are another couple of diagrams, one illustrating that the tangent used in trigonometry also means a tangent geometrically. Note that we have created a right-angled triangle. The next drawing exploits this information, and shows why both the angle and rise/run cannot both be proportioned equally. (I didn't follow the proof through to the bitter end, instead, I've left it for the interested reader to pursue).
There’s other useful information that can be extracted from the diagrams everyone has posted, but, I’ll save that for another time and stick to the point of the original question.
For anyone who’s tried to double, halve, add or subtract pitches and couldn’t make heads or tails of the results, there are also a couple of nifty formulas. Speaker: just out of curiosity, is this what you were trying to do in the first place that prompted your question?
Though the formulas are not directly related to the diagrams, they show the same thing; halving (or taking any other proportion of) a pitch does not halve an angle. For those who don’t like trig, the math can be done with pencil and paper and applied directly to a framing square. I’ve actually used the one example in the real world, dimensioning SIPS for a 4/12 shed roof on a 9/12 slope.
If anyone wants the proof of the tangent sum and difference formulas, there’s probably a zillion websites on the www showing how its done, or I can scan a textbook or my notes. Be careful what you ask for, you may get it! :twisted:

[Tom R]

Great stuff, - - highly technical, but also very interesting and informative.

Apparently you're not just the 'Average Joe'!

I'm sure we'll soon be needing some more of your expertise. Thanks.

[speaker]

Well I agree with Tom, Quite informative. thanks Mr bojANGLE. (Joe)
As for you curiosity, as to why I wanted to know. I read the infamous,

http://www.josephfusco.org/Articles/Roof_Cutting/raftercutting.htm

If you "edit - find" H.A.P. down the page some, you will come to his "H.A.P. Chart" on this I seen that a 6/12 was a 26.57 degree angle, and it was bothering me. I'll tell ya though, I didn't think that it was going to start this trig/tan freenzy.

hense "be careful what you ask for, you might get "

So, again I thank all for showing me the full understanding of this
(I feel I must find away to use this, so all these formulas weren't calc. in vain!) even though I also know everyone of you LOVED IT!

[speaker]

but further more "for the record I tried your calculations, and made it as far as

(18.43495) +(18.43495) (this part I get)
----------------------------
1-16/144 (this I don't)

[Joe Bartok]

I'll try to get this right the first time, but being a two-fisted rather than a mere two fingered typist, I'll likely have to come back and edit this.

The numerator in the fraction for that calculation is 4/12 + 4/12 = 8/12.

The term in the demominator, written out fully, is:
1 - (4*4)/(12*12) = 1 - 16/144 = 128/144. The terms in the numerators and denominators are multiplied.

Finally, solving the entire fraction:
(8/12)/(128/144) = (8*144)/(128*12) = .75, the same ratio as 9/12.

So, strange as it seems, the sum of two 4/12 pitches is a 9/12 pitch.

Yes, I had to do an edit. I tried typing (144*eight) the first time, and got this: (144*8)
Looks like I accidentally invoked the code for one of the "smilies." Gee, this math thing looks like it's going to be fun! Definitely have to stay on top of the proofreading.

[speaker]

Joe, what I was trying to say is I don't understand the calculation at all

1- 16/144 = 128/144

because ther are no brackets, I would come up with -0.1041666/144

this is probably basic multiplication of fractions, but I either never did this or forgot, I am not quite feeling like a calculas major or a brain surgeon at this point, just being honest, I don't get it.

I could sit here and say, oh ya Joe that's great, thanks. that way I learn nothing right!

[Rich]

(144/144)-(16/144)=128/144
:)

[speaker]

again quite obvious (in hind site)
thanks

[Dragon]

Hello.

I was wondering if someone could tell me why a

12/12 pitch roof is a 45 o angle

but a 6/12 pitch roof is 26.57 o angle?

thank (agaim)

For the same reason that a 24/12 isn't a 90 degree angle.

The answer is in the 12 horizontal. You literally can't get to 90 degrees and still have 3 sides.

[speaker]

Yet another way to solidify the point.

thanks dragon

[mtnblu]

I had to reply even if this is an old question that may have been answered. A 6/12 pitch means for ever 6 inches that goes up (the rise) the distance out (the run) will be 12. Therefore if you want to figure this out remember that all of the angles in the completed triangle must add up to 180 degrees. When you draw this triangle the pitch will be at 30 degees up, the next corner straight down will be 60 degrees and the last corner to complete the the triangle will be a 90 degree angle back to your starting point. I know because I draft by hand and basically you can pick up you 30, 60, triangle used for drafting and that is your 6/12 pitch. The 12/12 pitch means 12 up and 12 over...45 degree angle in the first corner, 45 degrees down and 90 degrees over to complete the triangle. That's the 45 degree triangle used for drafting.

[mjpliv]

A picture is worth a 1000 words