# Teach Me A little Trig Please?



## KennMacMoragh (Sep 16, 2008)

loneframer said:


> Just playin' Kenn. I have my calculator out and my mind is in the zone. Just exercising the old noodle brother.:thumbsup:


dang, I would have screwed up the whole bird house! :sneaky2:

I've never tried to figure out an octagon mathematically because I always knew I could go to my framing square if I ever needed it. But here it is http://mathworld.wolfram.com/Octagon.html, just need to factor it out to get _s_.

_r_ =


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## loneframer (Feb 13, 2009)

KennMacMoragh said:


> dang, I would have screwed up the whole bird house! :sneaky2:
> 
> I've never tried to figure out an octagon mathematically because I always knew I could go to my framing square if I ever needed it. But here it is http://mathworld.wolfram.com/Octagon.html, just need to factor it out to get _s_.
> 
> _r_ =


 See, there's my downfall. I know how to find out the answers I need, but to write out the formula... Well I think by using what you posted, I just made a batch of Plutonium.:laughing:


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## KennMacMoragh (Sep 16, 2008)

loneframer said:


> See, there's my downfall. I know how to find out the answers I need, but to write out the formula... Well I think by using what you posted, I just made a batch of Plutonium.:laughing:


Well, for sure there's more than one way to do it. I just ran that formula through my calculator, 6/(.5(1+√(2))) and got the same answer you did, 4.97, so they both work. But in my mind, a one line formula where you can just plug in the number is a lot simpler than this



loneframer said:


> Looking at it in plan view, a 22.5 degree angle, for every inch of common run, you will have 1.0823" of hip run and .4142" of fascia. That's a cheat sheet I made for myself.
> 
> Now, taking a 6" span, devided in half, leaves you a 3" common run, 3.2469" of hip run and 1.2426 " of facia from centerline, or 2.4852" overall, hip to hip.
> 
> ...


But hey, stick with what's easiest for ya :thumbsup:


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## loneframer (Feb 13, 2009)

KennMacMoragh said:


> Well, for sure there's more than one way to do it. I just ran that formula through my calculator, 6/(.5(1+√(2))) and got the same answer you did, 4.97, so they both work. But in my mind, a one line formula where you can just plug in the number is a lot simpler than this
> 
> 
> 
> But hey, stick with what's easiest for ya :thumbsup:


 Exactly.:thumbsup: Takes about two minutes to get every number I need to cut those pieces, still faster than stepping it off with a framing square.:laughing:


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## WNYcarpenter (Mar 2, 2007)

I've never been taught how to use a framing square for advanced roof framing...There's been a few guys on the crew who attempt to figure roofing with a calculator, but we always end up doing things the dummy way.

No one even knows how to 'Really' use a CM.....

We have some pretty slick tricks, but we've never done a clean precise framing project. I do maybe one roof framing project a year.

I hope I get enough time this weekend to study this thread until I can take this info and put it to use.


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## Warren (Feb 19, 2005)

I think an octagon can be achieved by using a series of 5/12 lines. 5/12 roof pitch is 22.5 degrees right? Works for me anyway.


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## loneframer (Feb 13, 2009)

Warren said:


> I think an octagon can be achieved by using a series of 5/12 lines. 5/12 roof pitch is 22.5 degrees right? Works for me anyway.


 22.5 degrees is actually 4.970/12, but yeah, close enough for most applications.:laughing:


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## Warren (Feb 19, 2005)

loneframer said:


> 22.5 degrees is actually 4.970/12, but yeah, close enough for most applications.:laughing:


The thread of your pencil is bound to be off that much.:thumbsup:


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## loneframer (Feb 13, 2009)

loneframer said:


> 22.5 degrees is actually 4.970/12, but yeah, close enough for most applications.:laughing:





Warren said:


> The thread of your pencil is bound to be off that much.:thumbsup:


 4.970/12 makes it very easy to determine the exact width of each octagon segment. 120" span = 60" run /12 = 5' * 4.970 = 24.85 (centerline) * 2 = 49.7" (width of segment). Using 5/12, the width would be 50". Over greater distances, it could create more drastic problems.


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## Warren (Feb 19, 2005)

loneframer said:


> 4.970/12 makes it very easy to determine the exact width of each octagon segment. 120" span = 60" run /12 = 5' * 4.970 = 24.85 (centerline) * 2 = 49.7" (width of segment). Using 5/12, the width would be 50". Over greater distances, it could create more drastic problems.


So on the 22" birdhouse were talking less than 1/8". I think we can deal with that. I would hope that on something much bigger I would have some specifications to go by.


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## rwa (May 6, 2009)

.... think I'd better stick with a flat roof :jester:


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## loneframer (Feb 13, 2009)

Warren said:


> So on the 22" birdhouse were talking less than 1/8". I think we can deal with that. I would hope that on something much bigger I would have some specifications to go by.


 No argument from me.:notworthy

I've been laughed at more than once for figuring roof components to the 4th decimal. The thing is, 3/100 of an inch is meaningless in and of itself. However, when figuring components on complex roofs with multiple planes/pitches and complex designs, that 3/100 could mean the difference of a 30' 3.5x12 microlam bastard hip fitting or falling short.:laughing:


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## Warren (Feb 19, 2005)

loneframer said:


> No argument from me.:notworthy
> 
> I've been laughed at more than once for figuring roof components to the 4th decimal. The thing is, 3/100 of an inch is meaningless in and of itself. However, when figuring components on complex roofs with multiple planes/pitches and complex designs, that 3/100 could mean the difference of a 30' 3.5x12 microlam bastard hip fitting or falling short.:laughing:


Preaching to the choir there Riz. I use a combination of math and science. Math to figure all the commons and common hips, and science to scientifically measure the expensive rafters prior to cutting them. I love the fact that you can figure every rafter on a complex roof ahead of time. I have worked on some real doozies and never learned to calculate all the complex fittings with a calculator. I consider myself a hybrid framer. I know all of the simple math, most of it by heart, and just enough complex math to keep me from getting too confident. In my world there is no substitute to measuring the actual length when its right there in front of you.

As far as octagons go, I don't ever remember a time when I needed to calculate one. Most times the foundation is there, and we just adjust as needed to make all the walls the same dimension. Whether they are exactly 45 degrees each is a moot point.


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## smeagol (May 13, 2008)

Yeah i cheat too and use 12 for the common 13 for the hip and 10 for the wall/facia:thumbsup:


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## loneframer (Feb 13, 2009)

My world has always been the contrary. More often than not, I have a series of poles and whalers, which are never parallel to the building line and rarely straight. Anything that gets built on top is paralleled off of the property line and most bays/octagonal components are either cantilevered or overframed on top of deck joists.

On the occasion that I get to frame off of a foundation, I always make sure the octagon is square with the building and true to itself, even if it means bumping it out on the corners of the foundation.

Most masons around here can't even get them close and I won't sacrifice my sanity trying to make them look good. If it's true, the pieces of the puzzle fit and if they fit, it makes it easy on everyone that follows. Especially when the GC is the guy behind me.:laughing:


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## Birch (Jul 20, 2009)

WNYcarpenter said:


> A tried to build an octagon roof for the BH competition with flat stock. I faked it by laying out a circle and bisecting it 4 times, connecting the dots then cutting each wedge at 22* bevel.. It was all f'ed up (I ended with 6 sides). I sanded and doctored it up and called it good enough
> 
> I have no idea what pitch I ended with, but I would have liked a 12/12.
> 
> ...






WNY,


This stuff is useful for just about any roof situation so I included Irregular solutions too. I hope I didn't convolute it up too much by doing so.


For finding rafter lengths quickly there is the Line Length Ratio method. For typical regular hip roofs you can use the ‘length Com/Hip Rafter per Foot Run’ tables found on the better framing squares. (For instance; under the 12 pitch column; 16.97” divided by 12 = 1.41416…. or the ^2. That is the Line Length Ratio for the common 12 pitch rafter. Now divide the Hip’s per foot dim by 12 to get the Hip’s LLR.)

For irregular hip angles (non 45 degree plan angles @ the hips) like you encounter with Irregular Hips or with octagons, hexagons, polygons, etc. shapes, you could do some basic full scale drawings and perform some basic math divisions to find the Line Length Ratios that will work for your particular roof. For irregular hip roofs you will do/solve sets of factors, or LLR’s, and for equal sided polygons you will just need one set.

For using a 4 x 8 sheet as a drawing board you may need to scale the plan and roof drawing down to fit. I usually pick a common run dimension of 4’ (major pitch, or lesser, for irregulars) Once you have the Hip’s plan view angle drawn on the sheet and intersecting the commons run line at the 4’ length you can draw the equal dimension rise lines for both the common(s) and hip rafters and measure them. Divide these rafter lengths by 48 to solve the LLR factors. When solving for irregulars, divide the minor common’s rafter length by the minor run to solve the LLR for those rafters and if needed you can find the Irregular Hip’s LLR as solved from the Minor’s Run too. (The Minor Run is the plan view ratio of the Major’s Run. For instance; using 6 as the major and 8 as the minor, The minor run is 6/8ths of 48” in the above explanation. That is also another LLR ratio too. The other plan view ratio is 8/6ths.)

Now for the best and most useful Trigk of all; divide each pitch’s Common Rafter length by the other pitches run to get a third (or fourth?) set of LLRs. These two last solved factors can be used to easily calculate Jack Rafter lengths from layouts. Just measure the layout from ‘Out’ of the jack’s layout to ‘In’ of the Irreg Hip and multiply that dimension by the appropriate LLR and that’s the jack length to the long point of the cheek bevel.

For solving the Hip’s plumb angle for marking with a Framing Square just divide the Hip’s Run by 4. (Pitch on Hip’s run/4)

Of course, you could just buy the CM Pro calculator and use it instead. :clap:


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## Joe Carola (Jun 15, 2004)

WNYcarpenter said:


> A tried to build an octagon roof for the BH competition with flat stock. I faked it by laying out a circle and bisecting it 4 times, connecting the dots then cutting each wedge at 22* bevel.. It was all f'ed up (I ended with 6 sides). I sanded and doctored it up and called it good enough
> 
> I have no idea what pitch I ended with, but I would have liked a 12/12.
> 
> ...


There are many ways to figure out Octagon sides, hip runs, hip lengths, sheathing cuts and backing bevels for an Octagon roof.

Plan view is always the easiest to work with. Your octagon hip runs at 67.5 degrees in plan view. When you have the width, figuring out the sides is very easy with a Construction Master or a Scientific calculator. A Construction Master with Trig is the best and easiest for me anyway. You have two calculators in one.

To figure out just your roof panels for example you can do this.


*Figuring the 8 sides for your 6" diameter.*

6 [Inch] [x] 22.5 [Tan] = 2.485281" (Length of 8 Sides)


*Or no Trig.*

67.5 [Pitch] 6 [Inch] [Rise] [Run] = 2.485281" (Length of 8 Sides and base of Triangle)


*12/12 Common Rafter length Trig.*

3 / 45 [Cos] = 4.242641" (Common Rafter Length and Triangle Height)


You have your complete roof panel with the height and base.


*The bevel at the hip would go like this.*

45 [Sine] x [67.5 [Cos] [=] [Conv] [Sine] = 15.69986 degrees.


For figuring out an Octagon roof, you still want to use plan view.

The framing square numbers are all based on 12" of run. That is why we hold the square 12/12 for the common plumbcut and 12/17 for the hip plumbcut because the hip has a diagonal run of 16.97056" or 17 because it runs at 45 degrees.

The Octagon hip runs at 67.5 degrees and that run is 12.98871" or 13". That is why for a 12/12 pitch common you always hold the framing square at 12/13 for the Octagon hip plumbcut.

I'll use a 12/12 pitch 12" common run with a 12" rise for example to show you the hip run and hip length and pitch.

*
67.5 [Pitch] (Plan View)

12 [Inch [Rise] (Plan View)

[Run] = 4.970563 (That's 22.5 degrees)

[Diag] = 12.98871" (Hip [Run Plan View)

[=] [Run]

12 [Inch] [Rise] (Common and Hip Rafter Rise)

[Diag] = 17.68351" (Hip Length)

[Pitch] 42.73 degrees (Hip Plumbcut Angle)*

There's alot you can do with a CM calculator. Viewing roof framing as triangles always helps.


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## Joe Carola (Jun 15, 2004)

WNY,

Here's a drawing of your project with all the numbers. I hope it makes sense.


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## Kent Whitten (Mar 24, 2006)

I knew you would show up soon Joe. I see the discussions you guys have over to JLC and it just boggles the mind sometimes. I stand in awe of the real math guys...you, Tim, Sim...and especially Joe Fusco, may he rest in peace. I was so sad to hear that he passed away. He contributed immensely to the framing world as do you.


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## Joe Carola (Jun 15, 2004)

framerman said:


> I knew you would show up soon Joe. I see the discussions you guys have over to JLC and it just boggles the mind sometimes. I stand in awe of the real math guys...you, Tim, Sim...and especially Joe Fusco, may he rest in peace. I was so sad to hear that he passed away. He contributed immensely to the framing world as do you.


Thank you Kent. I just enjoy these discussions and always want to learn more. Joe will be missed. We had alot of great roof framing discussions like this one.


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## world llc (Dec 9, 2008)

i am so confused....


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## loneframer (Feb 13, 2009)

world llc said:


> i am so confused....


 Yet, in most peoples minds, Framers are the bottom of the barrel in the construction world. Go figure. (pun intended):thumbsup:


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## world llc (Dec 9, 2008)

loneframer said:


> Yet, in most peoples minds, Framers are the bottom of the barrel in the construction world. Go figure. (pun intended):thumbsup:


 that's because most of you brilliant bastards are alcoholics and degenerates or out right psycho's!!:whistling

who else walks the ridge?:no:

some of the best guys i know are framers....


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## loneframer (Feb 13, 2009)

world llc said:


> that's because most of you brilliant bastards are alcoholics and degenerates or out right psycho's!!:whistling
> 
> who else walks the ridge?:no:
> 
> some of the best guys i know are framers....


 Then you have the freaks of nature. Casual drinker (a 12 pack is good for a couple weeks around here, if I even have it in the house), single father of 2 with equal parenting time, never been arrested, never lost my license, but I'll admit, I like being on the ridge. Great view from up there.:thumbsup:


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## world llc (Dec 9, 2008)

thoes are some great views up there!

i was just busting chops about the drunks, everyone knows that all trades have their share of talented degenerates, not just framers:laughing:


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## loneframer (Feb 13, 2009)

world llc said:


> thoes are some great views up there!
> 
> i was just busting chops about the drunks, everyone knows that all trades have their share of* talented degenerates*, not just framers:laughing:


 Uhhuhuhuh, thanks? Uhhuhuhuh.:blink:


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## Kent Whitten (Mar 24, 2006)

world llc said:


> who else walks the ridge?:no:
> 
> some of the best guys i know are framers....


I always liked the ridge. The first one to the top. King of the hill. Every home I built I stood at the highest part and enjoyed the view.


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## Joe Carola (Jun 15, 2004)

WNYcarpenter said:


> A tried to build an octagon roof for the BH competition with flat stock. I faked it by laying out a circle and bisecting it 4 times, connecting the dots then cutting each wedge at 22* bevel.. It was all f'ed up (I ended with 6 sides). I sanded and doctored it up and called it good enough
> 
> I have no idea what pitch I ended with, but I would have liked a 12/12.
> 
> ...


WNY,

If you wanted to do this for a hip roof with 45 degrees the answer to your sheathing angles are right there on your framing square for any pitch whether your framing a 6" span BH or a 10' span hip roof.

For 45° hip, the numbers are right there on the framing square.


For example a 12/12 pitch.


*Common Rafter Length Per Foot Run - 16.97"

Hip or Valley Rafter Length - 20.78"*

That's all based on a 12" run. You know in plan view that the run would be 12" in from both sides completing a perfect square.

With that 16.97" number or 17" on the corner of a sheet of plywood or big drawing paper you just come in from the left corner for example and make a mark at 12". 

Take your framing square and make a plumb mark from the 12" mark 16.97". 

Connect the 16.97" mark to the corner of the sheet of plywood and you now have your sheathing angle. It will measure 20.78" as it says on your framing square for your Hip Length.

For your 6" span BH roof, you make a 3" and 6" mark fro left to right.

Square up your 3" mark until it hits the 20.78" angled mark and connect that mark to the 6" mark and you now have your complete BH roof pattern.

Hope this makes sense. What the framing square numbers tell you right there is that it gives you the exact sheathing angle and complete little hip roof angle with a 12" run.

Just wanted to add that you can just do all this with the framing square and no tape.


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## Joe Carola (Jun 15, 2004)

Here it is on a sheet of plywood.


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## Joe Carola (Jun 15, 2004)

WNY,

Another way to get the sheathing angle using your framing square is to use the 12/12 Common Rafter Length Per Foot Run length of 16.97 and the 12" run number.

Start from the same corner on the left side of the plywood with the 12" mark on the framing square and slide the square on the right side until it reads 16.97". Scribe the 12" side and that's your angle.


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## Joe Carola (Jun 15, 2004)

Here's the drawing using the framing square.


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## KennMacMoragh (Sep 16, 2008)

Joe, just curious what CAD program you are using?


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## UpNorth (May 17, 2007)

Looks like Joe's trusty old Sketchup. The best free tool a carpenter ever had.


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## Joe Carola (Jun 15, 2004)

KennMacMoragh said:


> Joe, just curious what CAD program you are using?


It's Sketchup 7 free download.

http://sketchup.google.com/download/gsu.html


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