Hexagon and Octagon Roof Framing Geometry

Tait Chappell told me about the advance timber framing class at Fox Maple they put on last week where the students built a hexagon 13.5/12 common pitch with two dormers at 22/12 valleys joining into rotated purlin headers. So I decided to draw out a similar hexagon plan. I couldn't let Tait have all the fun.

The day I started to draw out  the hexagon roof plan my son Brian told me a funny, but sad story that needs addressing. He's been working for a company this last month that builds commercial buildings, about 25,000 sqft a year for the last 10 years. So they know what their doing, right??? We'll their current job, a prevailing wage job at $62.00 an hour, was bid too low so the owner of the company decided that he and his partner would cut and stack the octagon roof. Well, the owner cut the octagon hips as if it were a normal hip rafter and of course the hip rafters didn't fit. Then my son Brian explained to them that the octagon hip rafter rafters are not at the common rafter pitch and their not at the 90° eave angle hip rafter slope. He told them how to calculate the octagon hip rafters the old fashion way. Use the rise and the hip rafter run to calculate the octagon hip rafter slope angle. 

So both of them had Construction Master Calculators. So why is it that they couldn't figure out the octagon hip rafter slope angle? Their ignorance? Or the Construction Master Calculator? It's 2012, why in the world world anyone call their calculator a "Master" when it can't even calculate an octagon hip rafter that the Romans and Greeks did 2,000+ years ago. Or the French carpenters did  500+ years ago.

Here's a drawing showing how to calculate the hip rafter slope angle for an octagon roof the old fashion way using a CMC calculator. Or they could have purchased a Chappell Master Framing Square that has the Hexagon and Octagon roof framing information printed on the square.

Using the Chappell Master Framing Square, line 1 under 8:12, .6159 . Framing Square usage: 6.159 on the tongue and 10 on the body to draw out the plumb and level cuts of the octagon hip rafter. 

Or arctan(.6159) = 31.62892°

Using the Chappell Master Framing Square, line 2 under 8:12, 2.902 * 56" = 162.4854", it doesn't get any easier than that.

Using the Chappell Master Framing Square, line 3 under 8:12, 3.069 * 56" = 171.8648", it doesn't get any easier than that.

Here's the drawing I started working on. Hexagon roof with13.5: common rafter pitch and 22:12 dormers into a rotated purlin header. The hexagon was drawn out using Ad Triangulum ( equilateral triangle rotated within the circle). Just like the Romans and Greeks did 2,000+ years ago. Do the French carpenters of today use Ad Triangulum to draw out hexagons?

Rafter Tools is an android app with functions to calculate complex roof framing angles including Hexagon and Octagon roofs.