Prow Peak: Miter and Saw Blade Bevel Angles

Prow Peak: Miter and Saw Blade Bevel Angles

Over 12 Values of Angles

Common Pitch Angle = 36.86990° = 9"
90° - Common Pitch Angle = 53.13010° = 16"
Deck Angle = 65.00000° = 25 23/32"
90° - Deck Angle = 25.00000° = 5 19/32"
Prow Pitch Angle = 34.20516° = 8 5/32"
90° - Prow Pitch Angle = 55.79484° = 17 21/32"
Angle between "Ridge" and "Valley Trough" = 69.54216° = 32 5/32"
90° - Angle between "Ridge" and "Valley Trough" = 20.45784° = 4 15/32"
Backing Angle = 14.68892° = 3 5/32"

Prow Rafter Compound Angles

If the cuts at the peak and foot of the unbacked prow rafter are parallel:
Upper shoulder miter at peak and foot = 60.58422° = 21 9/32"
Blade Angle upper shoulder at peak and foot = 30.62973° = 7 3/32"
Blade Angle = 25.00000° along plumb line (34.20516° Miter angle on saw) or :
Blade Angle = 30.62973° along 60.58422° angle on upper shoulder (29.41578° Miter angle on saw)
The supplementary angles on the face created by cutting the compound angle:
90° - 16.02759° = 73.97241° and 90° + 16.02759° = 106.02759°
Over 12 : 16.02759° = 3 7/16"

If the compound angle as seen in plan at the prow rafter foot is at right angles rather than parallel to the cut at the peak:
The cuts at the foot on the unbacked prow rafter are:
Upper shoulder miter at foot = 21.08915° = 4 5/8"
Blade Angle upper shoulder at foot = 13.74364° = 2 15/16"
Blade Angle = 65.00000° along plumb line (34.20516° Miter angle on saw) or :
Blade Angle = 13.74364° along 21.08915° angle on upper shoulder (68.91085° Miter angle on saw)
The supplementary angles on the face created by cutting the compound angle:
90° - 31.63495° = 58.36505° and 90° + 31.63495° = 121.63495°
Over 12 : 31.63495° = 7 13/32"

Related Angles and Formulas

Differences in Length

Jack Rafters
Spacing O.C. along Prow Ridge × tan 69.54216°
Jack Purlins
Spacing O.C. along Common Rafter × tan 20.45784°

Spacing along Prow Rafter

Jack Rafters
Spacing O.C. along Prow Ridge ÷ cos 69.54216°
Jack Purlins
Spacing O.C. along Common Rafter ÷ cos 20.45784°

Sample Purlin Calculations
Purlins are spaced 3' - 9" (3.75' or 45") as viewed in plan

Calculating Difference in Length using the Deck Angle :
90° - Deck Angle = 25.00000°
Difference in Length
= Spacing O.C. in Plan × tan 25.00000°
= 45" × tan 25.00000°
= 20.98384" = 1' - 8 31/32"
Check : 4 × 20.98384" = 83.93536" = 83 15/16" = Prow Ridge

Calculations using the Angle on the Plane of the Roof (Angle between "Ridge" and "Valley Trough") :
90° - Angle between "Ridge" and "Valley Trough" = 20.45784°
Projection of Length in Plan to the Common Rafter
= 45" ÷ cos 36.86990° = 56.25"
Difference in Length
= Spacing O.C. on Common Rafter × tan 20.45784°
= 56.25" × tan 20.45784°
= 20.98385" = 1' - 8 31/32"
Spacing along Prow Rafter
= Spacing O.C. along Common Rafter ÷ cos 20.45784°
= 56.25" ÷ cos 20.45784°
= 60.03653" = 5' - 0 1/32"
Check : 4 × 60.03653" = 240.14612" = 240 5/32" = Prow Length

Compound Angles

Jack Rafters
65.00000° Blade Angle along Plumb Line (36.86990° Miter angle on saw) or :
14.68892° Blade Angle (Backing Angle) along 20.45784° angle on face set in the plane of the roof (69.54216° Miter angle on saw)
Either of these cuts produces supplementary angles:
90° - 34.20516° = 55.79484° and 90° + 34.20516° = 124.20516° on the Jack rafter face matching the side (plumb) face of the backed Prow Rafter.
Check : arccos (cos 53.13010° cos 20.45784°) = 55.79484°

Jack Purlins
19.76083° Blade Angle along angle 74.36920° on the face of the Purlin perpendicular to the plane of the roof (15.63080° Miter angle on saw) or :
14.68892° Blade Angle (Backing Angle) along 69.54216° angle on face set in the plane of the roof (20.45784° Miter angle on saw)
Either of these cuts produces supplementary angles:
90° - 5.40374° = 84.59626° and 90° + 5.40374° = 95.40374° on the Jack purlin face matching the side (plumb) face of the backed Prow Rafter.
Check : arccos (cos 74.36920° cos 69.54216°) = 84.59626°