If 12/12 rafters are cut on a 45 degree angle, aren’t 6/12 rafters cut on a 22-1/2 degree angle?
I’ve always assumed this to be true, and have never screwed up the cuts, thanks to using a framing square, but I thought I’d try the speed square yesterday, and it appears 22-1/2 lines up roughly with 5/12 pitch, and 6/12 is around 27-1/2 or so.
Can someone enlighten me why 22-1/2 degrees and 6/12 don’t match? (or do they)
edit: please read the thread before you reply…I’ve gotten the same (great) answers a number of times now! (24/12 isn’t 90 degrees, look at the protractor, use inverse tangent etc)
Edited 11/15/2005 8:18 am ET by Brian
Replies
It does if you use 6/19.2 thats what those diamonds are for on yer tape.
regards
Rik
no, withthe 19.2, thats 17. something degrees.. guess without calculator.
better check again
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rise over run ..... vs rise over length of rafter, so
e.g. invtan 12/12 = 45; invtan 6/12 = 26.565; invtan 4.97/12 = 22.5
sine 22.5 = .38 = 4.5922/12
Wow. That takes me back to high school. I loved that stuff then, but I have forgotten much of it. So I cannot think in simple ratios when I am cutting rafters. I really thought 22-1/2 was 6/12 pitch, and would have lost any number of bets.
So since sine 30 = .5 = 6/12 then 30 degrees is the cut?
I've used lots of algebra and geometry in the real world, and trigonometry was fun, but I'm rusty.
Now if the board and I were traveling at the speed of light...
edit: Whoops, now we are into inverse tangents, so forget the above calculations, I showed my work and still get an F
Edited 11/13/2005 3:26 pm ET by Brian
> So since sine 30 = .5 = 6/12 then 30 degrees is the cut?
Actually, rise over run is tangent. Sine would be rise over the distance on the rafter. Per Pythagoras, for a run of 12 and rise of 6, that's about 13.4164. InvTan(.5) is very close to 29.5 degrees, from there down to zero sine and tangent are quite close.
-- J.S.
"So since sine 30 = .5 = 6/12 then 30 degrees is the cut?"You don't want sine, you want tangent. Definition of TAN is rise/run.6/12 inv Tan is 26.33 degrees. Grab your speed square, check it out. If you call your 6/12 roof 26-1/2 degrees you're close enough for most work.
Did anyone read the edit at the bottom of my post?
Treat every person you meet like you will know them the rest of your life - you just might!
I did now <G>.
Brian, perhaps the attached diagram will help. The bisector of an angle does not bisect the rise/run.Joe Bartok
Brian,
I used to think the same until I realize an important point. The lower the pitch that you start with the great the difference you realize when increasing the numerator factor by 1. For instance the difference between 1/12 and 2/12 is much greater than the difference between 9/12 & 10/12.
Have you ever mistaken a 2/12 roof for a 4/12 roof? Probably not, although 14/12 might easily appear to be 16/12.
The reasons are discovered when you delve into the mathematics of it. When graphing the differential you get an asymptotic curve, which means it gets closer and closer to zero without ever getting there. Just like you can't divide a number by 2 and ever truly reach nothing.
Another (semi) easy way to think about it is that if 12/12 (45°) is twice the pitch of 6/12, than 24/12 would be plumb because it's 2x a 45° angle.
Jon Blakemore
RappahannockINC.com Fredericksburg, VA
That was a fantastic explaination Jon. Seriously. I knew the answer, but never could have put it into words like that.... nicely done.
Wow.I'm glad you understood it.You know that feeling right after you hit the post button and you think "I'm just an idiot, why didn't I wait for somebody like Joe Fusco to explain that", well I had that feeling.
Jon Blakemore RappahannockINC.com Fredericksburg, VA
Well, you had me at "asymptotic curve". ;)
Right on. I was with an engineer friend tonight and he basically told me the same asymptotic deal - but your 24/12 explanation really seals the deal.
Its a good thing I don't design trusses...
Treat every person you meet like you will know them the rest of your life - you just might!
Another way to look at Jon's explanation is to look at a protractor. If you trace a line around the circumference of the protractor from 0 degrees to 45 degrees you are going up the y axis (that would be vertically on a roof) faster than you are moving across the x axis. From 45 degrees to 90 degrees you start moving across the x axis faster than you move up the y axis. Same explanation, different example.
That's an illustration of why as Jon said you would not confuse a 2/12 for a 4/12 but a 16/12 wouldn't look much differnt than an 18/12 from the ground.
Also why you can't express a vertical surface as a pitch -- it's undefined.
Brian if you take that notion further, wouldn't 24/12 be 90°?
The relationship isn't a linear one, it's a sine curve. As you approach exactly vertical or horizontal, the rise/run ratios become extreme. The standard 1% slope we use on most exterior grades could be expressed as 0.125/12, and of course absolute vertical can't be expressed at all by this system since there is infinite rise.