Roof Framing Angle Formulas
Formulas for the Rise assume a Run = 12
Definitions Examples Diagrams
Framing and Joinery Angle Calculator
 W = Angle viewed in plan between Hip eaves or Valley ridges; any value greater than zero up to 180°. Common Rafter Pitch Angle or Common Rafter Plumb Cut Angle arctan (Pitch) or arctan (Rise ÷ Run) The initial values of Pitch Angles are generally given as Over 12 Angle in Plan or Deck Angle measured between the Hip eave and the Hip run, or the ridge viewed in plan and the Valley run. Also the Cheek Cut Angle or Saw Blade Bevel Angle along the plumb lines of Jack Rafters, at the Hip peak or the Valley foot. Major Side Plan Angle = arctan [sin W ÷ (Major Pitch / Minor Pitch + cos W)] Major Side Plan Rise = 12 × [sin W ÷ (Major Pitch / Minor Pitch + cos W)] Minor Side Plan Angle = arctan [sin W ÷ (Minor Pitch / Major Pitch + cos W)] Minor Side Plan Rise = 12 × [sin W ÷ (Minor Pitch / Major Pitch + cos W)] Hip-Valley Rafter Pitch Angle or Hip-Valley Rafter Plumb Cut Angle Angle = arctan (tan Common Rafter Pitch Angle × sin Plan Angle) Rise = 12 × (tan Common Rafter Pitch Angle × sin Plan Angle) Jack Rafter Side Cut Angle on the face of the Jack Rafter 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) Sheathing Angle, also the angle on a Jack Purlin on the face of the Purlin set in the plane of the roof. The Sheathing Angle is 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) Backing Angle. 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 (also 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) Plumb Backing Angle. Backing Angle 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) Hip-Valley Rafter Side Cut Angle at the Hip Rafter foot or Valley Rafter peak. Created on the upper shoulder 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) Hip-Valley Rafter Side Cut Angle at the Hip Rafter peak or Valley Rafter foot. Created on the upper shoulder of the Hip-Valley Rafter as the rafter intercepts a plumb plane at right angles the Hip or Valley eave. Angle = arctan (cos Hip-Valley Rafter Pitch Angle ÷ tan Plan Angle) Rise = 12 × (cos Hip-Valley Rafter Pitch Angle ÷ tan Plan Angle) Formula for the Saw Blade Bevel Angle to cut along the Hip-Valley Rafter Side Cut Angles: 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) 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) Purlin Saw Blade Bevel 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. Saw Blade Bevel Angle for the Hip-Valley Rafter to accommodate Square Cut 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) Square Cut Fascia Saw Miter Angle along the side or plumb face of the Hip-Valley Rafter. Purlins or Bird-Block also create this angle where they intercept the side or plumb face of a 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)