The first development below models a section of a Hip roof, with Deck angle DD on a level plane passing through the eaves, and the angle on the plane of the roof 90  P2 located at the Hip eave or base. The same development serves for a Valley rafter, but the deck is defined as a level plane passing through the ridges, therefore DD and 90  P2 are located at the Valley ridge or peak. The R4 angles also follow this pattern: R4B angles are projections of DD and D at the Hip rafter eave or Valley rafter ridge. R4P angles are projections of 90  DD and 90  D at the Hip rafter peak or Valley rafter foot. 
Development of Main Side Angles:
Common Pitch angle = SS Hip Pitch angle = R1 Deck angle = DD Angle on the Roof Plane = P2 The unit length for the Hip run and the trigonometric lengths given for the line segments on the drawings may be disregarded. The developments may be drawn to any scale, using only a straightedge and a compass for the line segments marked equal. Substitute angles S for SS, and D for DD, to develop the Adjacent side angles. 
Development of Backing Angles (C5):
W = DD + D, where 0 < W < 180 degrees Produce the lines defining the Hip eaves or Valley ridges Construct the triangle of Hip Pitch angle R1 Construct line segment sin R1 Rotate line segment sin R1 to the line of the Hip run produced Construct the line segments tan DD and tan D perpendicular to the Hip run Draw the line segments (hypotenuse) for C5 angles tan C5m = sin R1 ÷ tan DD tan C5a = sin R1 ÷ tan D 
Development of Angles on the Bottom Face of the Valley
at the Hip Eaves, or Valley Ridges (R4B): W = DD + D, where 0 < W < 180 degrees Produce the lines defining the Hip eaves or Valley ridges Construct the triangle of Hip Pitch angle R1 Rotate line segment 1/cos R1 to the Hip run Construct the line segments tan DD and tan D perpendicular to the Hip run Construct the parallel line segments tan DD and tan D perpendicular to the Hip run produced to 1/cos R1 Draw the line segments (hypotenuse) for R4B angles tan R4Bm = tan DD / (1/cos R1) = cos R1 tan DD tan R4Ba = tan D / (1/cos R1) = cos R1 tan D 
Development of Angles on the Bottom Face of the Valley
at the Hip Peaks, or Valley Feet (R4P): W = DD + D, where 0 < W < 180 degrees Produce the lines defining 180  W = 90  DD + 90  D Construct the triangle of Hip Pitch angle R1 Rotate line segment 1/cos R1 to the Hip run Construct the line segments tan (90  DD) and tan (90  D) perpendicular to the Hip run Construct the parallel line segments tan (90  DD) and tan (90  D) perpendicular to the Hip run produced to 1/cos R1 Draw the line segments (hypotenuse) for R4P angles tan R4Pm = (1/tan DD) / (1/cos R1) = cos R1 / tan DD tan R4Pa = (1/tan D) / (1/cos R1) = cos R1 / tan D 
Formulas for Hip/Valley Joinery at Eaves and Ridges:
The dihedral angle, or angle between two planes, where the bottom face of a Hip or Valley rafter intercepts plumb planes passing through the eaves or ridges, is defined as angle 90 + A5 or 90  A5. Therefore, the saw blade angle setting along Miter line R4 on the Valley, or along a line following R5 on a plumb face, is A5. The Bevel line on the plumb Valley face is 90  R1. tan Blade angle = sin Miter / tan Bevel tan A5B = sin R4B / tan (90  R1) = sin R4B tan R1 tan A5P = sin R4P / tan (90  R1) = sin R4P tan R1 Angles 90 + R5 and 90  R5 are created on the cross section of the Valley rafter by cutting the compound angle. cos Angle on Compound Face = cos Miter cos Bevel cos (90  R5B) = cos R4B cos (90  R1) sin R5B = cos R4B sin R1 cos (90  R5P) = cos R4P cos (90  R1) sin R5P = cos R4P sin R1 
Valley rafter foot meets plumb planes at walls, showing the major planes and angles.
