W : Angle viewed in plan between Hip eaves or Valley ridges; any value greater than zero up to 180°.

SS : Main or Major side Common Rafter Pitch Angle or Common Rafter Plumb Cut Angle
S : Adjoining or Minor side Common Rafter Pitch Angle or Common Rafter Plumb Cut Angle
Angle = arctan (Pitch) or arctan (Rise ÷ Run)
Rise = 12 × tan Common Pitch Angle or 12 × Rise ÷ Run
The initial values of the Common Rafter Pitches are generally given as Over 12

DD : Main or Major side Cheek Cut Angle or Saw Blade Bevel Angle to cut along the plumb lines on a Jack Rafter, at the Hip peak or the Valley foot. Angle in Plan or Deck Angle measured between the Main or Major side Hip eave and the Hip run, or the Main or Major side ridge viewed in plan and the Valley run.
D : Adjoining or Minor side Cheek Cut Angle or Saw Blade Bevel Angle to cut along the plumb lines on a Jack Rafter, at the Hip peak or the Valley foot. Angle in Plan or Deck Angle measured between the Adjoining or Minor side Hip eave and the Hip run, or the Adjoining or Minor side ridge viewed in plan and the Valley run.
Main Side Plan Angle = arctan [sin W ÷ (Main Pitch / Adjoining Pitch + cos W)]
Main Side Plan Rise = 12 × sin W ÷ (Main Pitch / Adjoining Pitch + cos W)
Adjoining Side Plan Angle = arctan [sin W ÷ (Adjoining Pitch / Main Pitch + cos W)]
Adjoining Side Plan Rise = 12 × sin W ÷ (Adjoining Pitch / Main Pitch + cos W)

R1 : Hip-Valley Rafter Pitch Angle or Hip-Valley Rafter Plumb Cut Angle. The intercept of the Jack Rafter with the Hip-Valley Rafter creates the footprint angles 90° ± R1 on the side (plumb) face of the Hip-Valley Rafter.
Angle = arctan (tan Common Rafter Pitch Angle × sin Plan Angle)
Rise = 12 × tan Common Rafter Pitch Angle × sin Plan Angle

P2 : Jack Rafter Side Cut Angle, located on the Jack Rafter face set in the plane of the roof.
Angle = arctan (cos Common Rafter Pitch Angle ÷ tan Plan Angle)
Rise = 12 × cos Common Rafter Pitch Angle ÷ tan Plan Angle

90° – P2 : Sheathing Angle, also the angle on a Jack Purlin on the face set in the plane of the roof. Complementary to the Jack Rafter Side Cut Angle.
Angle = arctan (tan Plan Angle ÷ cos Common Rafter Pitch Angle)
Rise = 12 × tan Plan Angle ÷ cos Common Rafter Pitch Angle

C5 : Backing Angle, the Saw Blade Bevel Angle to cut a ridge on a Hip Rafter or trough on a Valley Rafter. Also the Saw Blade Bevel Angle when cutting along the Sheathing Angle, the angle on the face of the Purlin set in the plane of the roof (the Sheathing Angle) or the complementary Jack Rafter Side Cut Angle.
Angle = arctan (sin Hip-Valley Rafter Pitch Angle ÷ tan Plan Angle)
Rise = 12 × sin Hip-Valley Rafter Pitch Angle ÷ tan Plan Angle

A7 : Plumb Backing Angle. Backing Angle C5 projected to or viewed along the Hip-Valley Rafter Plumb Cut.
Angle = arctan (tan Hip-Valley Rafter Pitch Angle ÷ tan Plan Angle)
Rise = 12 × tan Hip-Valley Rafter Pitch Angle ÷ tan Plan Angle

R4B : Hip-Valley Rafter Side Cut Angle at the Hip Rafter foot or Valley Rafter peak. Angle on the stick created on the upper/lower shoulders of the Hip-Valley Rafter as the rafter intercepts a plumb plane following the Hip eave or Valley ridge line. This angle may occur at the Valley Rafter foot if the plumb plane intercepted follows the eave line.
Angle = arctan (cos Hip-Valley Rafter Pitch Angle × tan Plan Angle)
Rise = 12 × cos Hip-Valley Rafter Pitch Angle × tan Plan Angle

R4P : Hip-Valley Rafter Side Cut Angle at the Hip Rafter peak or Valley Rafter foot. Angle on the stick created on the upper/lower shoulders of the Hip-Valley Rafter as the rafter intercepts a plumb plane at right angles the Hip or Valley eave line.
Angle = arctan (cos Hip-Valley Rafter Pitch Angle ÷ tan Plan Angle)
Rise = 12 × cos Hip-Valley Rafter Pitch Angle ÷ tan Plan Angle

A5B : Saw Blade Bevel Angle to make the cut along the Hip-Valley Rafter Side Cut Angle R4B at the Hip Rafter foot or Valley Rafter peak.
A5P : Saw Blade Bevel Angle to make the cut along the Hip-Valley Rafter Side Cut Angle R4P at the Hip Rafter peak or Valley Rafter foot.
Angle = arctan (sin Hip-Valley Rafter Side Cut Angle × tan Hip-Valley Rafter Pitch Angle)
Rise = 12 × sin Hip-Valley Rafter Side Cut Angle × tan Hip-Valley Rafter Pitch Angle

R5B : Hip-Valley Rafter Footprint Angle or Mortise Layout Angle , 90° ± R5B is created by cutting the Compound Angle defined by R4B and 90° – R1 at the intercept of the Hip-Valley Rafter and a plumb plane following the Hip eave or Valley ridge line.
Angle = arctan (cos Plan Angle × tan Hip-Valley Rafter Pitch Angle)
Rise = 12 × cos Plan Angle × tan Hip-Valley Rafter Pitch Angle

R5P : Hip-Valley Rafter Footprint Angle or Mortise Layout Angle , 90° ± R5P is created by cutting the Compound Angle defined by R4P and 90° – R1 at the intercept of the Hip-Valley Rafter and a plumb plane at right angles to the eave line.
Angle = arctan (sin Plan Angle × tan Hip-Valley Rafter Pitch Angle)
Rise = 12 × sin Plan Angle × tan Hip-Valley Rafter Pitch Angle

P1 : Purlin Saw Miter Angle on the faces of Purlins or Bird-Block set perpendicular to the plane of the roof, at the intercept of this face with the side or plumb face of the Hip-Valley Rafter.
Angle = arctan (sin Common Rafter Pitch Angle ÷ tan Plan Angle)
Rise = 12 × sin Common Rafter Pitch Angle ÷ tan Plan Angle

R2 : Square Tail Fascia Saw Miter Angle on the Hip-Valley Rafter. Purlins or Bird-Block intercepting the side or plumb face of a Hip-Valley Rafter create the footprint angles 90° ± R2 on the Hip-Valley Rafter.
Angle = arctan (sin Backing Angle × tan Jack Rafter Side Cut Angle)
Rise = 12 × sin Backing Angle × tan Jack Rafter Side Cut Angle

C1 : Purlin Saw Blade Bevel Angle to make the cut along 90° – P1 on the faces of Purlins or Bird-Block set perpendicular to the plane of the roof, at the intercept of this face with the side or plumb face of the Hip-Valley Rafter. Also the Saw Blade Bevel Angle along the miter line of angle R2 on the Hip-Valley Rafter to accommodate Square Tail Fascia.
Angle = arctan (sin Purlin Saw Miter Angle ÷ tan Common Rafter Pitch Angle)
Rise = 12 × sin Purlin Saw Miter Angle ÷ tan Common Rafter Pitch Angle

90° – R3 : Angle on the stick, located on the upper/lower shoulders of a Hip-Valley Rafter. 90° – R3 is the Hip Rafter Square Tail Fascia Side Cut Angle at the foot of the Hip Rafter, or the Valley Rafter intercepts Main Purlin Side Cut Angle at the peak of the Valley Rafter.
Angle = arctan (tan Sheathing Angle ÷ cos Backing Angle)
Rise = 12 × tan Sheathing Angle ÷ cos Backing Angle

C2 : Saw Blade Bevel Angle to make the cut along the miter line of angle 90° – R3 on the upper/lower shoulders of the Hip-Valley Rafter.
Angle = arctan (cos Hip Rafter Square Tail Fascia Side Cut Angle × tan Backing Angle)
Rise = 12 × cos Hip Rafter Square Tail Fascia Side Cut Angle × tan Backing Angle

SFMm : Main Side Saw Miter Angle for Square Tail Fascia
Main Side Angle = arctan (sin Main Common Pitch Angle ÷ tan Adjoining Plan Angle)
Major Side Rise = 12 × sin Main Common Pitch Angle ÷ tan Adjoining Plan Angle

SFMa : Adjoining Side Saw Miter Angle for Square Tail Fascia
Minor Side Angle = arctan (sin Adjoining Common Pitch Angle ÷ tan Main Plan Angle)
Minor Side Rise = 12 × sin Adjoining Common Pitch Angle ÷ tan Main Plan Angle

SFB: Saw Blade Bevel Angle for Square Tail Fascia
Angle = arctan (cos Square Tail Fascia Miter Angle × tan Jack Rafter Side Cut Angle)
Rise = 12 × cos Square Tail Fascia Miter Angle × tan Jack Rafter Side Cut Angle

NOTE : For the following "Layover" formulas the intersection of Main and Adjoining (Dormer) Ridge Lines as viewed in Plan is assumed to be 90°
Refer to the Framing Angle Calculators and diagrams to solve compound angles for the general case where W is any value.

PLM: Saw Miter Angle for Purlin intersects Main Roof (Layover)
Angle = arctan (cos Adjoining Common Pitch Angle ÷ tan Main Common Pitch Angle)
Rise = 12 × cos Adjoining Common Pitch Angle ÷ tan Main Common Pitch Angle

PLB: Saw Blade Bevel Angle for Purlin intersects Main Roof (Layover)
Angle = arcsin (cos Main Common Pitch Angle × sin Adjoining Common Pitch Angle)

PLC: Footprint Angle or Mortise Layout Angle on Purlin face abutting the slope of the Main Roof
Angle = arctan [tan SS ÷ (cos SS × sin S × cos S)]
Rise = 12 × tan SS ÷ (cos SS × sin S × cos S)