Hip/Valley Rafter Backing Angle Chart in decimal inches

Eave Angle = 90
Plan Angle = 45
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 10.02 9.87 10.02 0.13 0.13 0.13 0.26 0.27 0.27 0.27 0.28 0.28 0.30 0.31 0.31 0.45 0.45 0.45 0.48 0.49 0.49
4 18.43 13.26 12.92 13.26 0.17 0.18 0.18 0.34 0.35 0.35 0.36 0.37 0.37 0.40 0.41 0.41 0.59 0.60 0.60 0.63 0.65 0.65
5 22.62 16.42 15.78 16.42 0.21 0.22 0.22 0.42 0.44 0.44 0.44 0.46 0.46 0.49 0.52 0.52 0.72 0.75 0.75 0.78 0.81 0.81
6 26.57 19.47 18.43 19.47 0.25 0.27 0.27 0.50 0.53 0.53 0.52 0.55 0.55 0.58 0.62 0.62 0.85 0.91 0.91 0.92 0.97 0.97
7 30.26 22.42 20.87 22.42 0.29 0.31 0.31 0.57 0.62 0.62 0.60 0.64 0.64 0.67 0.72 0.72 0.98 1.06 1.06 1.05 1.13 1.13
8 33.69 25.24 23.09 25.24 0.32 0.35 0.35 0.64 0.71 0.71 0.67 0.74 0.74 0.75 0.82 0.82 1.09 1.21 1.21 1.17 1.30 1.30
9 36.87 27.94 25.10 27.94 0.35 0.40 0.40 0.70 0.80 0.80 0.73 0.83 0.83 0.82 0.93 0.93 1.20 1.36 1.36 1.29 1.46 1.46
10 39.81 30.51 26.92 30.51 0.38 0.44 0.44 0.76 0.88 0.88 0.79 0.92 0.92 0.89 1.03 1.03 1.30 1.51 1.51 1.40 1.62 1.62
11 42.51 32.95 28.54 32.95 0.41 0.49 0.49 0.82 0.97 0.97 0.85 1.01 1.01 0.95 1.13 1.13 1.39 1.66 1.66 1.50 1.78 1.78
12 45.00 35.26 30.00 35.26 0.43 0.53 0.53 0.87 1.06 1.06 0.90 1.10 1.10 1.01 1.24 1.24 1.48 1.81 1.81 1.59 1.94 1.94
13 47.29 37.45 31.30 37.45 0.46 0.57 0.57 0.91 1.15 1.15 0.95 1.20 1.20 1.06 1.34 1.34 1.56 1.96 1.96 1.67 2.11 2.11
14 49.40 39.52 32.47 39.52 0.48 0.62 0.62 0.95 1.24 1.24 0.99 1.29 1.29 1.11 1.44 1.44 1.63 2.11 2.11 1.75 2.27 2.27
15 51.34 41.47 33.52 41.47 0.50 0.66 0.66 0.99 1.33 1.33 1.03 1.38 1.38 1.16 1.55 1.55 1.70 2.26 2.26 1.82 2.43 2.43
16 53.13 43.31 34.45 43.31 0.51 0.71 0.71 1.03 1.41 1.41 1.07 1.47 1.47 1.20 1.65 1.65 1.76 2.42 2.42 1.89 2.59 2.59

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 10.02 9.87 10.02 3/16 3/16 3/16 5/16 5/16 5/16 5/16 5/16 5/16 5/16 5/16 5/16 1/2 1/2 1/2 1/2 1/2 1/2
4 18.43 13.26 12.92 13.26 3/16 3/16 3/16 3/8 3/8 3/8 3/8 3/8 3/8 7/16 7/16 7/16 5/8 5/8 5/8 11/16 11/16 11/16
5 22.62 16.42 15.78 16.42 1/4 1/4 1/4 7/16 1/2 1/2 1/2 1/2 1/2 1/2 9/16 9/16 3/4 13/16 13/16 13/16 13/16 13/16
6 26.57 19.47 18.43 19.47 5/16 5/16 5/16 9/16 9/16 9/16 9/16 9/16 9/16 5/8 5/8 5/8 7/8 15/16 15/16 15/16 15/16 15/16
7 30.26 22.42 20.87 22.42 5/16 5/16 5/16 5/8 5/8 5/8 5/8 11/16 11/16 11/16 3/4 3/4 15/16 1 1/16 1 1/16 1 1/16 1 3/16 1 3/16
8 33.69 25.24 23.09 25.24 3/8 3/8 3/8 11/16 3/4 3/4 11/16 3/4 3/4 3/4 7/8 7/8 1 1/8 1 1/4 1 1/4 1 3/16 1 5/16 1 5/16
9 36.87 27.94 25.10 27.94 3/8 7/16 7/16 3/4 13/16 13/16 3/4 7/8 7/8 7/8 15/16 15/16 1 1/4 1 3/8 1 3/8 1 5/16 1 1/2 1 1/2
10 39.81 30.51 26.92 30.51 7/16 1/2 1/2 13/16 15/16 15/16 13/16 15/16 15/16 15/16 1 1/16 1 1/16 1 5/16 1 9/16 1 9/16 1 7/16 1 5/8 1 5/8
11 42.51 32.95 28.54 32.95 7/16 1/2 1/2 7/8 15/16 15/16 7/8 1 1/16 1 1/16 15/16 1 3/16 1 3/16 1 7/16 1 11/16 1 11/16 1 1/2 1 13/16 1 13/16
12 45.00 35.26 30.00 35.26 7/16 9/16 9/16 7/8 1 1/16 1 1/16 15/16 1 1/8 1 1/8 1 1/16 1 1/4 1 1/4 1 1/2 1 13/16 1 13/16 1 5/8 1 15/16 1 15/16
13 47.29 37.45 31.30 37.45 1/2 5/8 5/8 15/16 1 3/16 1 3/16 15/16 1 1/4 1 1/4 1 1/8 1 3/8 1 3/8 1 9/16 1 15/16 1 15/16 1 11/16 2 1/8 2 1/8
14 49.40 39.52 32.47 39.52 1/2 5/8 5/8 15/16 1 1/4 1 1/4 1 1 5/16 1 5/16 1 1/8 1 1/2 1 1/2 1 11/16 2 1/8 2 1/8 1 13/16 2 5/16 2 5/16
15 51.34 41.47 33.52 41.47 1/2 11/16 11/16 1 1 3/8 1 3/8 1 1/16 1 7/16 1 7/16 1 3/16 1 9/16 1 9/16 1 3/4 2 5/16 2 5/16 1 7/8 2 7/16 2 7/16
16 53.13 43.31 34.45 43.31 9/16 3/4 3/4 1 1/16 1 7/16 1 7/16 1 1/8 1 1/2 1 1/2 1 1/4 1 11/16 1 11/16 1 13/16 2 7/16 2 7/16 1 15/16 2 5/8 2 5/8

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