Exterior Rake Crown Moulding Angles jig

Thomas Clark asked me for some help with the miter and bevel angles on an 8:12 pitched roof with a crown spring angle of 38° for the exterior rake crown moulding. After doing this study on the exterior rake crown miter and bevel angles  for the crown angles I should have asked him for help. Right from the start he said the reference plane for the roof angles for the crown moulding were 52° and 90° - ( 38° - 33.69007°) = 85.69007°. I don't know why I didn't use these two angles from the start, but it turns out these are the correct angles to find the miter angles of the exterior rake crown molding.

I made a test jig for a roof with a slope angle of 33.69007° and used a spring angle of 33.69007°. This model was to be the control model. With these settings the miter angle on the eave rake crown is 33.69007° and the miter angle on the rake crown is 90°. Using a saw blade bevel angle of 45° the bevel angle on the eave crown moulding  resulted in a  39.76216, which is the same as P2, jack rafter side cut angle.

Test jig with crown blocks.

Test jig turned upside down, so I could use the top of the workbench as the roof surface plane.

2x4 test cuts placed in the position of the crown molding.

Back to the original crown molding problem.  Just when I thought I had a handle on the crown angles, the miter angles can now be calculated using the reference plane, but I still need a formula for the  bevel angles on the top edge of the crown molding.  When I built the test jig with the crown blocks nailed to the side of the eave and rake lines I thought the model was more of a study for Square Tail Fascia angles. But after I turned the test jig upside down these crown angle are more of a study for hopper angle. Hoppers are just roofs turned upside down. So using the roof sheathing angles for the miter angle on the side of the crown molding makes sense. 

In this next drawing I was able to dimension the bevel angles on top of the 2x4's in the drawing  and using a saw blade bevel angle formula I calculated the saw blade bevel angles for each side.

Saw Blade Bevel Angle From Miter Line Side = arctan(sin(Miter Angle on Stick) ÷ tan(Bevel Angle on Stick)
Saw Blade Bevel Angle From Miter Line Side = arctan(sin(52.07869°) ÷ tan(39.56508°) = 43.67381°

Saw Blade Bevel Angle From Miter Line Side = arctan(sin(86.60130°) ÷ tan(46.27535°) = 43.67427°

Formula for calculating the miter angles on the face of the crown molding.
Face Miter Angle = arctan( cos ( Reference Plane ) ÷ tan ( Plan Angle ))
Eave Molding Face Miter Angle = 90° - arctan( cos (85.69007 ) ÷ tan ( 5.50984 )) = 52.07869
Rake Molding Face Miter Angle = 90° - arctan( cos ( 52 ) ÷ tan ( 84.49016 )) = 86.60130

Formula for calculating the width of the rake crown molding
Rake Crown Width = Eave Crown Width ÷ cos( spring angle)
Rake Crown Width = 4" ÷ cos( 38°) = 5.07607"

Another way of developing the width of the rake crown molding.

Got around to making the test jig for the 8:12 roof with the crown spring angle of 38°. The 43.67° saw blade bevel angle wasn't perfect. However, it might have been my bevel setting on my compound miter saw. Following the miter intersection of the two pieces on the roof surface produced an angle of 37.5° and it should have been 39.5°. This discrepancy might be due to the saw blade bevel angle not being set perfectly on the compound miter saw. But the miters look pretty good. The length of the rake crown is 5.07", 5 1/16", and the width of the eave crown is 4". The two pieces were a perfect match for the length of the corner miter.

Yikes!!! More roof surface angles. Now this study of the rake crown moulding angles is looking like a prow roof study.

Gary Katz's Website with examples of rake crown moulding.

Like Gary said on his page the square cut rafter tails were intentional. The square cut rafter tails could have been cut at 90°, but they also cut have been cut to 85.60097° for use with the 38°/52° standard crown moulding profiles. The rafter tails cut at 85.60097° would be like the crown pitch blocks I installed on my model.

Cutting Raked Crown Moulding with a Handsaw.