Hip/Valley Rafter Backing Angle Chart in decimal inches

Eave Angle = 135
Plan Angle = 67.5
Pitch Angle = arctan ( Pitch 12)
Hip Pitch Angle = arctan( tan (Pitch Angle) * sin ( Plan Angle))
Hip Backing Angle = arctan ( sin ( Hip Pitch Angle) tan ( Plan Angle) )
Hip Backing Angle Rise = ( Material Width * .5 ) * tan ( Hip Backing Angle) ) or Valley Trough Depth
Plan Angle Unit Rise = tan ( Plan Angle )
Hip Unit Rise Run = 12 sin ( Plan Angle )
Unit Hip Drop = Pitch ( Hip Unit Rise Run * (Plan Angle Unit Rise * 2 )
Hip Drop Adjustment = ( Material Width * .5 ) * Unit Hip Drop
Plumb Backing Angle = tan (Hip Pitch Angle) tan (Plan Angle)
Plumb Backing Angle Rise = Backing Angle Rise cos Hip Pitch Angle
Pitch Pitch Angle Hip Angle Hip Backing Angle Hip Plumb Backing Angle 2x
rise  drop  plumb
3x
rise  drop  plumb
3 1/8
rise  drop  plumb
4x
rise  drop  plumb
5 1/8
rise  drop  plumb
6x
rise  drop  plumb
3 14.04 13.01 5.33 5.46 0.07 0.07 0.07 0.14 0.14 0.14 0.15 0.15 0.15 0.16 0.17 0.17 0.24 0.25 0.25 0.26 0.26 0.26
4 18.43 17.12 6.95 7.27 0.09 0.10 0.10 0.18 0.19 0.19 0.19 0.20 0.20 0.21 0.22 0.22 0.31 0.33 0.33 0.34 0.35 0.35
5 22.62 21.05 8.46 9.06 0.11 0.12 0.12 0.22 0.24 0.24 0.23 0.25 0.25 0.26 0.28 0.28 0.38 0.41 0.41 0.41 0.44 0.44
6 26.57 24.79 9.85 10.83 0.13 0.14 0.14 0.26 0.29 0.29 0.27 0.30 0.30 0.30 0.33 0.33 0.45 0.49 0.49 0.48 0.53 0.53
7 30.26 28.32 11.12 12.58 0.15 0.17 0.17 0.29 0.33 0.33 0.31 0.35 0.35 0.34 0.39 0.39 0.50 0.57 0.57 0.54 0.61 0.61
8 33.69 31.63 12.26 14.31 0.16 0.19 0.19 0.33 0.38 0.38 0.34 0.40 0.40 0.38 0.45 0.45 0.56 0.65 0.65 0.60 0.70 0.70
9 36.87 34.72 13.27 16.01 0.18 0.22 0.22 0.35 0.43 0.43 0.37 0.45 0.45 0.41 0.50 0.50 0.60 0.74 0.74 0.65 0.79 0.79
10 39.81 37.59 14.18 17.69 0.19 0.24 0.24 0.38 0.48 0.48 0.39 0.50 0.50 0.44 0.56 0.56 0.65 0.82 0.82 0.69 0.88 0.88
11 42.51 40.26 14.99 19.33 0.20 0.26 0.26 0.40 0.53 0.53 0.42 0.55 0.55 0.47 0.61 0.61 0.69 0.90 0.90 0.74 0.96 0.96
12 45.00 42.73 15.70 20.94 0.21 0.29 0.29 0.42 0.57 0.57 0.44 0.60 0.60 0.49 0.67 0.67 0.72 0.98 0.98 0.77 1.05 1.05
13 47.29 45.02 16.33 22.52 0.22 0.31 0.31 0.44 0.62 0.62 0.46 0.65 0.65 0.51 0.73 0.73 0.75 1.06 1.06 0.81 1.14 1.14
14 49.40 47.15 16.89 24.06 0.23 0.33 0.33 0.46 0.67 0.67 0.47 0.70 0.70 0.53 0.78 0.78 0.78 1.14 1.14 0.84 1.23 1.23
15 51.34 49.11 17.39 25.56 0.23 0.36 0.36 0.47 0.72 0.72 0.49 0.75 0.75 0.55 0.84 0.84 0.80 1.23 1.23 0.86 1.32 1.32
16 53.13 50.93 17.83 27.03 0.24 0.38 0.38 0.48 0.77 0.77 0.50 0.80 0.80 0.56 0.89 0.89 0.82 1.31 1.31 0.88 1.40 1.40

Hip/Valley Rafter Backing Angle Chart in feet & inches

Pitch Pitch Angle Hip Angle Hip Backing Angle Hip Plumb Backing Angle 2x
rise  drop  plumb
3x
rise  drop  plumb
3 1/8
rise  drop  plumb
4x
rise  drop  plumb
5 1/8
rise  drop  plumb
6x
rise  drop  plumb
3 14.04 13.01 5.33 5.46 1/8 1/8 1/8 3/16 3/16 3/16 3/16 3/16 3/16 3/16 3/16 3/16 1/4 1/4 1/4 5/16 5/16 5/16
4 18.43 17.12 6.95 7.27 1/8 1/8 1/8 3/16 1/4 1/4 1/4 1/4 1/4 1/4 1/4 1/4 5/16 3/8 3/8 3/8 3/8 3/8
5 22.62 21.05 8.46 9.06 1/8 1/8 1/8 1/4 1/4 1/4 1/4 1/4 1/4 5/16 5/16 5/16 7/16 7/16 7/16 7/16 1/2 1/2
6 26.57 24.79 9.85 10.83 3/16 3/16 3/16 5/16 5/16 5/16 5/16 5/16 5/16 5/16 3/8 3/8 1/2 1/2 1/2 1/2 9/16 9/16
7 30.26 28.32 11.12 12.58 3/16 3/16 3/16 5/16 3/8 3/8 5/16 3/8 3/8 3/8 7/16 7/16 9/16 5/8 5/8 9/16 5/8 5/8
8 33.69 31.63 12.26 14.31 3/16 1/4 1/4 3/8 7/16 7/16 3/8 7/16 7/16 7/16 1/2 1/2 9/16 11/16 11/16 5/8 3/4 3/4
9 36.87 34.72 13.27 16.01 3/16 1/4 1/4 3/8 7/16 7/16 3/8 1/2 1/2 7/16 9/16 9/16 5/8 3/4 3/4 11/16 13/16 13/16
10 39.81 37.59 14.18 17.69 1/4 1/4 1/4 7/16 1/2 1/2 7/16 1/2 1/2 1/2 9/16 9/16 11/16 7/8 7/8 3/4 15/16 15/16
11 42.51 40.26 14.99 19.33 1/4 5/16 5/16 7/16 9/16 9/16 7/16 9/16 9/16 1/2 5/8 5/8 11/16 15/16 15/16 3/4 15/16 15/16
12 45.00 42.73 15.70 20.94 1/4 5/16 5/16 7/16 5/8 5/8 1/2 5/8 5/8 1/2 11/16 11/16 3/4 1 1 13/16 1 1/16 1 1/16
13 47.29 45.02 16.33 22.52 1/4 5/16 5/16 1/2 5/8 5/8 1/2 11/16 11/16 9/16 3/4 3/4 13/16 1 1/16 1 1/16 13/16 1 3/16 1 3/16
14 49.40 47.15 16.89 24.06 1/4 3/8 3/8 1/2 11/16 11/16 1/2 3/4 3/4 9/16 13/16 13/16 13/16 1 3/16 1 3/16 7/8 1 1/4 1 1/4
15 51.34 49.11 17.39 25.56 1/4 3/8 3/8 1/2 3/4 3/4 1/2 3/4 3/4 9/16 7/8 7/8 13/16 1 1/4 1 1/4 7/8 1 3/8 1 3/8
16 53.13 50.93 17.83 27.03 1/4 7/16 7/16 1/2 13/16 13/16 9/16 13/16 13/16 5/8 15/16 15/16 7/8 1 5/16 1 5/16 15/16 1 7/16 1 7/16

Revolution of the Cutting Plane by Joe Bartok
Hip Drop, Mark & Back Bevel Angle by Joe Fusco
Hip and Valley Backing Angles