Hip and Valley Roof Systems
Geometric Models and Angle Formulas

 The Plan Angle is measured from the Hip Roof eave to the Hip run. Hip-Valley Pitch Angle = arctan (tan Common Pitch Angle × sin Plan Angle) Backing Angle = arctan (sin Hip-Valley Pitch Angle ÷ tan Plan Angle)
 The Plan Angle is measured from a line at right angles to the Valley Roof eave to the Valley run. Note that given the same combination of Common Pitches as the Hip Roof the Plan Angle produced is complementary to that of the Hip Roof. The Plan Angle trig functions and operations must be corrected to reflect this change in location. For example: Hip-Valley Pitch Angle = arctan (tan Common Pitch Angle × cos Plan Angle) Backing Angle = arctan (sin Hip-Valley Pitch Angle × tan Plan Angle)
 The Plan Angle is measured from the Hip Roof eave to the Hip run or the Valley Roof ridge projected to plan and the Valley run. The resulting formulas are the same for both Hips and Valleys. Hip-Valley Pitch Angle = arctan (tan Common Pitch Angle × sin Plan Angle) Backing Angle = arctan (sin Hip-Valley Pitch Angle ÷ tan Plan Angle) Offsetting the convenience of having only one set of equations for both Hip and Valleys, the locations of many angles are affected since they "follow" the Plan Angle. For example: Hip Sheathing Angle: Measured from the Hip Roof eave to the Hip ridgeline. Valley Sheathing Angle: Measured from the Valley Roof ridge to the Valley trough. Hip-Valley Side Cut Angles: Angles at the Hip rafter foot are located at the Valley rafter ridge, those at the Hip rafter peak are at the Valley rafter foot. Other angles, for example angles required for cutting Purlins, are unaffected and have the same location and orientation on a Hip or Valley Roof.