Using the ground plan of a pentagon and traditional layout geometry for the bevel angles on the hip rafters at the peak of the hip rafter everything works out as expected. The geometry matches the roof framing trigonometric ratios as well.
Platonic Solid ICOSAHEDRON
Deck: Pentagon
Roof Surface Faces: 5 x Equilateral Triangles
Deck Angle = 108°
Plan DD = 54.00°
Common Rafter Slope Angle SS = 37.37737°
Hip Rafter Slope Angle R1 = 31.71747°
Dihedral Angle = 138.16038°
Hip Rafter Backing Angle = 20.91981°
Platonic Solid Edges
Hip Rafter Miter Angle = 31.71747°
Hip Rafter Bevel Angle = 58.2825°
Hip Rafter Saw Blade Bevel Angle = 54.00°
Roof Eave Angle = 108.00000 SS = Main Rafter Slope Angle = 37.37737 S = Adjacent Rafter Slope Angle = 37.37737 DD = Main Plan Angle = 54.00000 D = Adjacent Plan Angle = 54.00000 R1 = Hip Rafter Slope Angle = 31.71748 C5m Main Hip Rafter Backing Angle = 20.90516 C5a Adjacent Hip Rafter Backing Angle = 20.90516 P2m = Main Jack Rafter Side Cut Angle = 30.00000 90° - P2m = Main Roof Sheathing Angle = 60.00000 P2a = Adjacent Jack Rafter Side Cut Angle = 30.00000 90° - P2a = Adjacent Roof Sheathing Angle = 60.00000
R4 = 31.71747°R5B = 19.96463° A5B = 25.17126° R5P = 26.56505° A5P = 18.00000° R4B = 49.49929° R4P = 31.71747°
With a square ground plan the geometry matches the roof framing trigonometric ratios.
Roof Eave Angle = 90.00000 SS = Main Rafter Slope Angle = 31.74747 S = Adjacent Rafter Slope Angle = 31.74747 DD = Main Plan Angle = 45.00000 D = Adjacent Plan Angle = 45.00000 R1 = Hip Rafter Slope Angle = 23.63071 C5m Main Hip Rafter Backing Angle = 21.84290 C5a Adjacent Hip Rafter Backing Angle = 21.84290 P2m = Main Jack Rafter Side Cut Angle = 40.37702 90° - P2m = Main Roof Sheathing Angle = 49.62298 P2a = Adjacent Jack Rafter Side Cut Angle = 40.37702 90° - P2a = Adjacent Roof Sheathing Angle = 49.62298
R4 = 42.49429°
R5B = 17.19096° A5B = 16.46543° R5P = 17.19096° A5P = 16.46543° R4B = 42.49429° R4P = 42.49429°
Here's another square ground plan, but with unequal roof slope angles.
Hip Rafter A
Roof Eave Angle = 90.00000 SS = Main Rafter Slope Angle = 45.00000 S = Adjacent Rafter Slope Angle = 60.00000 DD = Main Plan Angle = 60.00000 D = Adjacent Plan Angle = 30.00000 R1 = Hip Rafter Slope Angle = 40.89339 C5m Main Hip Rafter Backing Angle = 20.70481 C5a Adjacent Hip Rafter Backing Angle = 48.59038 P2m = Main Jack Rafter Side Cut Angle = 22.20765 90° - P2m = Main Roof Sheathing Angle = 67.79235 P2a = Adjacent Jack Rafter Side Cut Angle = 40.89339 90° - P2a = Adjacent Roof Sheathing Angle = 49.10661
Main
P4 = 19.28633°
R5B = 23.41322° A5B = 34.53758° R5P = 36.86990° A5P = 19.10661° R4B = 52.62876° correct
R4P = 23.57818° ... wrong
Adjacent
R4 = 52.62876°
R5B = 36.86990° A5B = 19.10661° R5P = 23.41322° A5P = 34.53758° R4B = 23.57818° ... correct R4P = 52.62876° ... wrong
From the 3D drawing and the geometry
R4Pm = 38.27204°
R4Pa = 22.75094°
Update
I need a way of changing
R1 = Hip Rafter Slope Angle B = 35.26439
to
R1 = Hip Rafter Slope Angle B = 34.35489
_______________________________________________________________
Roof Eave Angle = 105.00000 SS = Main Rafter Slope Angle = 40.89339 S = Adjacent Rafter Slope Angle = 34.35489 DD = Main Plan Angle = 43.77559 D = Adjacent Plan Angle = 61.22441 R1 = Hip Rafter Slope Angle = 30.92781 C5m Main Hip Rafter Backing Angle = 28.20944 C5a Adjacent Hip Rafter Backing Angle = 15.76254 Valley Sleeper Saw Blade Bevel Angle = 90° - (C5m + C5a) = 46.02802 P2m = Main Jack Rafter Side Cut Angle = 38.27161 90° - P2m = Main Roof Sheathing Angle = 51.72839 P2a = Adjacent Jack Rafter Side Cut Angle = 24.38937 90° - P2a = Adjacent Roof Sheathing Angle = 65.61063
Hip Rafter A side Cut Angle R4P = P2m
Hip Rafter B side Cut Angle R4P = P2a
_____________________________________________________________________
Hip Rafter B
Roof Eave Angle = 90.00000 SS = Main Rafter Slope Angle = 45.00000 S = Adjacent Rafter Slope Angle = 45.00000 DD = Main Plan Angle = 45.00000 D = Adjacent Plan Angle = 45.00000 R1 = Hip Rafter Slope Angle = 35.26439 C5m Main Hip Rafter Backing Angle = 30.00000 C5a Adjacent Hip Rafter Backing Angle = 30.00000 P2m = Main Jack Rafter Side Cut Angle = 35.26439 90° - P2m = Main Roof Sheathing Angle = 54.73561 P2a = Adjacent Jack Rafter Side Cut Angle = 35.26439 90° - P2a = Adjacent Roof Sheathing Angle = 54.73561
R4 = 39.23152°
R5B = 26.56505° A5B = 24.09484° R5P = 26.56505° A5P = 24.09484° R4B = 39.23152°... correct R4P = 39.23152°... wrong
R4P = 24.15203°... from drawing
Hip Rafter C
Roof Eave Angle = 90.00000 SS = Main Rafter Slope Angle = 45.00000 S = Adjacent Rafter Slope Angle = 60.00000 DD = Main Plan Angle = 60.00000 D = Adjacent Plan Angle = 30.00000 R1 = Hip Rafter Slope Angle = 40.89339 C5m Main Hip Rafter Backing Angle = 20.70481 C5a Adjacent Hip Rafter Backing Angle = 48.59038 P2m = Main Jack Rafter Side Cut Angle = 22.20765 90° - P2m = Main Roof Sheathing Angle = 67.79235 P2a = Adjacent Jack Rafter Side Cut Angle = 40.89339 90° - P2a = Adjacent Roof Sheathing Angle = 49.10661
Main
P4 = 19.28633°
R5B = 23.41322° A5B = 34.53758° R5P = 36.86990° A5P = 19.10661° R4B = 52.62876° correct
R4P = 23.57818° ... wrong
Adjacent
R4 = 52.62876°
R5B = 36.86990° A5B = 19.10661° R5P = 23.41322° A5P = 34.53758° R4B = 23.57818° ... correct R4P = 52.62876° ... wrong
From the 3D drawing and the geometry
R4Pm = 38.27204°
R4Pa = 22.75094°
Hip Rafter D
Roof Eave Angle = 90.00000 SS = Main Rafter Slope Angle = 60.00000 S = Adjacent Rafter Slope Angle = 60.00000 DD = Main Plan Angle = 45.00000 D = Adjacent Plan Angle = 45.00000 R1 = Hip Rafter Slope Angle = 50.76848 C5m Main Hip Rafter Backing Angle = 37.76124 C5a Adjacent Hip Rafter Backing Angle = 37.76124 P2m = Main Jack Rafter Side Cut Angle = 26.56505 90° - P2m = Main Roof Sheathing Angle = 63.43495 P2a = Adjacent Jack Rafter Side Cut Angle = 26.56505 90° - P2a = Adjacent Roof Sheathing Angle = 63.43495
R4 = 32.31153°
R5B = 40.89339° ... slope angle of adjoining hip rafter A5B = 33.21091° R5P = 40.89339° ... slope angle of adjoining hip rafter A5P = 33.21091° R4B = 32.31153° ... correct R4P = 32.31153° ... wrong
From the 3D drawing and the geometryR4Pm = 68.45626°R4Pa = 68.45626°
Golden Rhombus Parallelogram
Hip Rafter A
Roof Eave Angle = 63.43490 SS = Main Rafter Slope Angle = 31.71750 S = Adjacent Rafter Slope Angle = 31.71750 DD = Main Plan Angle = 31.71745 D = Adjacent Plan Angle = 31.71745 R1 = Hip Rafter Slope Angle = 18.0000 C5m Main Hip Rafter Backing Angle = 26.56508 C5a Adjacent Hip Rafter Backing Angle = 26.56508 P2m = Main Jack Rafter Side Cut Angle = 54.0000 90° - P2m = Main Roof Sheathing Angle = 36.00 P2a = Adjacent Jack Rafter Side Cut Angle = 54.0000 90° - P2a = Adjacent Roof Sheathing Angle = 36.00
R5B = 15.45045° A5B = 9.34970° R5P = 9.69372° A5P = 15.24017° R4B = 30.44636°... correct R4P = 56.98263°... wrong
From the 3D drawing and traditional layout geometry
R4P = 43.56300°
Hip Rafter B
Roof Eave Angle = 116.56510 SS = Main Rafter Slope Angle = 31.71750 S = Adjacent Rafter Slope Angle = 31.71750 DD = Main Plan Angle = 58.28255 D = Adjacent Plan Angle = 58.28255 R1 = Hip Rafter Slope Angle = 27.73233 C5m Main Hip Rafter Backing Angle = 16.04506 C5a Adjacent Hip Rafter Backing Angle = 16.04506 P2m = Main Jack Rafter Side Cut Angle = 27.73227 90° - P2m = Main Roof Sheathing Angle = 62.26773 P2a = Adjacent Jack Rafter Side Cut Angle = 27.73227 90° - P2a = Adjacent Roof Sheathing Angle = 62.26773
R5B = 15.45045° A5B = 23.31858° R5P = 24.09488° A5P = 14.16082° R4B = 55.07574°... correct R4P = 28.68045° ... wrong
From the 3D drawing and traditional layout geometry
R4P = 41.51304°
Joe Bartok's pentahedron models .
Development of the Pentahedron Extracted from the Peak of the Hip Rafter
The gap in between the models of the pentahedron is the reason the hip rafter head cuts don't work. The plumb hip rafter head cut angles or the bevel angles on top of the hip rafters need to be changed.
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