Hip Rafter Backing Angles for Edge Bevel

I've written an a couple of articles on how to draw the dihedral angle triangle for the hip/valley rafter backing angles, but after reading an JLC article it needs to be clarified again. So no one makes the mistake of using the hip rafter plumb backing angle instead of the hip rafter backing angle to edge bevel the hip/valley rafter.

Here's a wire frame sketch showing the location of the hip/valley rafter backing angle and the hip/valley rafter plumb backing angle.

The hip/valley rafter plumb backing angle is plumb to the earth, or plumb to the ground plan.

The dihedral angle triangle for the hip/valley rafter backing angles is perpendicular to the hip/valley rafter in elevation. You only use the hip rafter backing angle to edge bevel hip/valley rafters. 

Hip/Valley Rafters are backed out to align with the roof planes.

How to draw the hip/valley rafter backing angle in plan view.

How to draw the hip/valley rafter backing angle in section view.

For equal pitched roofs .

Here are all the hip/valley rafter backing angle formulas(C5) by Joe Bartok.

• Hip Rafter Backing Angle  = arccos (sin Plan Angle ÷ cos Jack Rafter Side Cut Angle)
• Hip Rafter Backing Angle  = arccos (sin Plan Angle ÷ sin Sheathing Angle)
• Hip Rafter Backing Angle  = arccos (cos Common Rafter Pitch Angle ÷ cos Hip-Valley Pitch Angle)
• Hip Rafter Backing Angle  = arcsin (sin Common Rafter Pitch Angle x cos Plan Angle)
• Hip Rafter Backing Angle  = arctan (tan Common Rafter Pitch Angle x sin Jack Rafter Side Cut Angle)
• Hip Rafter Backing Angle  = arctan (tan Common Rafter Pitch Angle x cos Sheathing Angle)
• Hip Rafter Backing Angle  = arcsin (tan Jack Rafter Side Cut Angle x tan Hip-Valley Pitch Angle)
• Hip Rafter Backing Angle  = arcsin (tan Hip-Valley Pitch Angle ÷ tan Sheathing Angle)
• Hip Rafter Backing Angle  = arctan (sin Hip-Valley Pitch Angle ÷ tan Plan Angle)
• Hip Rafter Backing Angle  = arctan (sin Hip-Valley Pitch Angle x tan Adjacent Plan Angle)

Here are some the hip/valley rafter plumb backing angle formulas (A7) by Joe Bartok.

• Hip Rafter Plumb Backing Angle = arctan (tan(Hip Rafter Slope Angle) x sin(Hip Rafter Slope Angle))
• Hip Rafter Plumb Backing Angle = arctan( tan( Common Rafter Slope Angle ) * cos( Plan Angle ))

I only know of 2 types of roof framing that you would use the hip rafter plumb backing angle to cut material related to a roof. 

1. Hip Rafter Post rotated perpendicular to the hip rafter run line.
2. Hip Rafter Rake Wall Studs.

Here's an study by Joe Bartok on the angle A7, when the Hip Rafter Post is rotated perpendicular to the hip rafter run line.

Here's a study I did on the angle A7, when the Hip Rafter Post is rotated perpendicular to the hip rafter run line.

You can also use the hip rafter plumb backing angle for hip rafter rake walls rotated into the roof surface plane. The rake stud is cut using the hip rafter plumb backing angle as the miter angle and the formula to calculate the saw blade bevel angle to cut the compound cut of the hip rafter rake wall stud uses A7.

Saw blade bevel angle = P5BV = arctan(sin A7 x tan DD)