Roofing Angles

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Getting the Rafter Triangle | getting top cuts | getting side cuts | hip / purlin side angle

Getting the Rafter Triangle

Most angles can be learned from a basic rectangular hip roof.  When the hips are not equal, or the plan has irregular angles, the same principles apply.

I have to thank Jack in Connecticut for his correspondence on the subject and his information on the steel square etc.  Good on you mate.  You got me motivated!

For the purposes of this exercise let's forget about overhangs at the eaves.

hip roof calculation
Roofing angles - plan and elevations of a hip roof.

Let's say that I am given the plan, with 2400 as my span of the roof.  I am also given a couple of elevations.  The slope of the roof is given in the US style of the slope ratio which is 18 in 12.  I will use this example as a roof that is 24ft x 48ft.

It could just as easily be 2400 mm x 4800 mm  and using Aussie manner of degrees. This particular pitch of 56 and a bit degrees is hard to draw, so I would have to use a good protractor or a calculator.  The 18 in 12 couldn't be easier.

steel square
Roofing angles - Using a steel square with a couple of battens.
rafter top cut
Roofing angles - the main roof triangle
  • I set up a couple of stools and drop a half sheet of fairly clean MDF on top.
  • I go to a nice square end  and put a mark 36" along from the corner.
  • I put a mark 24" up the  edge and join them with a straight edge and pencil.
  • Straight away I can set up my steel square with a couple of battens screwed to it that I use for my main rafter bevels.
  • If the square gets kicked of dropped I just check it against the triangle on my setting out piece of MDF.
  • Looking at my plan, I see that the span is 24'00 so my half span in 12'00.
  • I have just drawn my roof triangle to a scale of 2 inches to a foot :-)
  • 12'00 half span at  ratio of 18 to 12 gives me an 18'00 rise.  
  • I take my steel ruler and measure the length of the other side.  That's the one that I penciled in wearing my glasses and with a sharp pencil and a straightedge.
  • I get a reading of 21.63 or thereabouts. 

So by drawing it out at a fairly big scale I can get very close to the exact size.  I have my main roof angles and my roof surface length.

Drawing it out like this is not new. For centuries people have been building roofs with extreme accuracy by just drawing it out to scale.  If you have the room you could draw it out full size, nothing wrong with that.  You can also do it with a calculator but for all practical purposes what's the point of  getting a figure of 21.633307653 when the practicalities of cutting and fixing on site say that you will be lucky if you get it within 1/4 of an inch.

As an aside, just before typing that last bit, I went to accessories and clicked on the calculator.  In the view menu I selected scientific and did the old Pythagoras trick.  Square root of 12 x12 + 18 x18.  = 21.633307653  Easy I could have got a lot more decimal places if I had wanted too.  I never knock calculators, they are great, but I like to have a manual back up to check to see if I made a mistake.

back to top

Developing the surfaces and getting the top cuts

Back to business, with that triangle and the plan we have all they information that we need to solve most of the problems in this roof.

Imagine that you had on the table in front of you a cardboard model of this roof. The four pieces joined at the ridge.  From the top the angles at the hips would all look at 45 deg, as in the normal plan drawing.

Now imagine pressing down on the ridge and flattening the model.  What you would see is this.

roof developement
Roofing angles - All the roof surfaces laid flat.

The eaves and the ridge lines have not changed.  The only thing that has changed is that the true length of the roof surface, 2163 has been used.  This is know as developing the surface.  As all the roof surfaces are the same length from the ridge to the eaves (2160), and the hips are the same length I only need to draw out one of these roof shapes at my scale on my MDF drawing board to get lots more information.

roof developed surface
Roofing angles - Detail of one surface.

Here I get into using quite technical terms, but I will try to explain them as I go along.

Let me just say this now, we are not talking about just carpentry work, the structural steel worker, the sheet metal roofer etc can adapt this information to their particular needs.

  • I already have the true lengths of the ridge and fascia, now I can measure the true length of the hips.
  • I can get the roof surface area.  21.63 x (48 - 12). for say a roof tiler or sheeter.

I could pencil in vertical lines and get the:-

  • The number of common rafters and their lengths. (common rafters are the full length one that fix to the ridge.
  • The jack rafters and their lengths (JL) and the jack rafter diminish.(JRD) (jack rafters are the ones that fix to the hip.  The diminish is the amount by which each jack rafter gets shorter than the one before.
  • Jack rafter top cut.
  • Hip top cut.
  • For the roofer I could get the number of sheets and the sheet length.
  • The sheet lengths,diminish at the hip and the angle of cut of the hip sheets.
roofing cuts and bevels
Roofing angles - Purlin top cuts.

I could pencil in lines parallel to the eaves/fascia and get:-

The number and lengths of the roof battens, or purlins.  Battens is the more usual term for tile roofs and purlins are used to fix the roofing sheets to.

Same angle for steel ties, under purlins and the end of the hips.

back to top

Getting the side cuts

There are a few bevels that we haven't seen yet.  I will use a rectangular hip roof again as an example.

developement of the hip triangle
Roofing angles - laying flat or developing the hip triangle.

If the roof were a solid, and I could slice vertically down through the ridge and hips I could make something like the above sketch. Where the four sections are separated at the hips I can see the hip triangles. It is easy enough to draw one out as though it was flat.  I have a right angle triangle with a known height and a known run.

Let's call one of the smaller four sided figures the hip solid.

purlin in the hip end.
Roofing angles - The hip solid

Now if that same hip solid had a huge purlin in it, I would see something like the sketch on the right.  The tops of purlins are in line with the roof surface, so the sides of the purlins are square off the roof surface.  That is at 90 deg. to it.

So if I sliced off a section (the green bit) square off the surface of the roof, underneath it I would see another triangle with my purlin side cuts on it.

Purlin side cuts.

getting purlin side cuts
Roofing angles - the purlin side cuts.

So here it is laid out as If on a sheet of MDF, using a tee square and compasses etc


The point that I made earlier still stands, all the work you see here can be drawn at a scale of 2" to a foot on a bit of ply with just a square, a ruler, a straight edge and pencil.

I get my distances A, B and C and lay them out on the plan view of the hip end.

It is just as though I have laid down the section through the roof at the green line flat.

This way of drawing it out, and then emphasising the angle with a marker pen and naming it always reminds me what angle I am looking at.

Some roofs have very similar angles, not a lot of difference in them, and if you don't make it clear which is which you could easily make a mistake.

to get the hip top cut
Roofing angles - Hip top cut

Here I do the same to get the hip triangle.  

I have changed to a lot steeper pitch to show the difference in the angles clearly.  

I arrange the green line that sections the hip solid at the point of the  right angle of the main triangle.

I get the hip triangle laid flat and immediately draw in the hip top cut.

Hip / purlin side angle

There is one more bevel that I can get out of the hip triangle. It is a line that is drawn on the hip to show the angle that a purlin makes when it touches the hip.

I only came up against this one when I started do structural steel roofs.   The first one I built, I got wrong and put my cleats square off the hip.  What a bummer!   I had to cut my cleats off and re weld them on again. Good job it was small job and easy to do.

Here is the photo from the steel roof section of the main roof page again to show you what I mean.

hips and purlins
Roofing angles - Steel purlin to hip connection.

hip with cleats
Roofing angles - marking out a steel hip for purlin cleats.

The steel cleats are about 6mm thick and punched for the purlin bolts.

They are welded to the RHS hip.  The "C" purlins are then bolted to the cleats.

A cheap way to get a large roof.

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Please Note! The information on this site is offered as a guide only!  When we are talking about areas where building regulations or safety regulations could exist,the information here could be wrong for your area.  It could be out of date!  Regulations breed faster than rabbits!
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