intercepts Plane of Main Roof

The geometry of the model is identical to the geometry of a Hip Roof.
The angle labelled Hip Pitch Angle in the diagram actually lies on the plane of the 7/12 Main Roof.
Solution of the Saw Blade Bevel
The Dormer Pitch Angle of 39.80557° occupies the same position on the model
as the Common Rafter Pitch Angle on a Hip Roof.
The Main Roof Pitch Angle of 30.25644° occupies the same position on the model
as the Plan Angle on a Hip Roof.
The Saw Blade Bevel Angle occupies the same position on the model
as the Backing Angle on a Hip Roof.
The solution given in the drawing substitutes the model ratios for the trig functions of the angles in the well-known formula: Backing Angle = arctan (sin Hip Pitch Angle ÷ tan Plan Angle)
But the Backing Angle for a Hip Rafter may also be calculated directly from
the Common Rafter Pitch Angle and Plan Angle by:
Backing Angle = arcsin (sin Common Rafter Pitch Angle × cos Plan Angle)
Noting the positions of the angles on the model we make the following substitutions: Saw Blade Bevel Angle = arcsin (sin Dormer Pitch Angle × cos Main Roof Pitch Angle)
= arcsin (sin 39.80557° × cos 30.25644°) = 33.57154°
Solution of the Saw Miter Angle and Angle on the Fascia
The Dormer Pitch Angle of 39.80557° occupies the same position on the model
as the Common Rafter Pitch Angle on a Hip Roof.
The Main Roof Pitch Angle of 30.25644° occupies the same position on the model
as the Plan Angle on a Hip Roof.
The Saw Miter Angle occupies the same position on the model
as the Jack Rafter Side Cut Angle on a Hip Roof.
Jack Rafter Side Cut Angle = arctan (cos Common Rafter Pitch Angle ÷ tan Plan Angle)
Noting the positions of the angles on the model we make the following substitutions: Saw Miter Angle = arctan (cos Dormer Pitch Angle ÷ tan Main Roof Pitch Angle)
= arctan (cos 39.80557° ÷ tan 30.25644°) = 52.78951° The complementary Angle on the Fascia is located in the same position on the model
as the Sheathing Angle on a Hip Roof.
Sheathing Angle = arctan (tan Plan Angle ÷ cos Common Rafter Pitch Angle)
Noting the positions of the angles on the model we make the following substitutions: Angle on the Fascia = arctan (tan Main Roof Pitch Angle ÷ cos Dormer Pitch Angle)
= arctan (tan 30.25644° ÷ cos 39.80557°) = 37.21049° |