Calculating with respect to a level plane through the Ridges
The Cricket Valley Roof Plane is assumed to be 22.5°, half of the 45° on the 12 ÷ 12 Slope.
The ridge of the 22.5° slope meets the 7 ÷ 12 ridge (30.25644° Pitch Angle) at 90° as measured on a Level Plane.
LAG Program Entries :
(Set Select Entry Mode to Angular)
Main Pitch : 30.25644°
Adjacent Pitch : 22.5°
Total Deck Angle : 90°
LAG Program Returns :
R1 = Cricket Valley Pitch Angle = 18.66136° : 4-1/16 ÷ 12 Pitch
DD = 7 ÷ 12 Side Deck Angle = 35.37781°
90° - P2m = Angle between 7 ÷ 12 Ridge and Cricket Valley Trough = 39.42235°
C5m = 7 ÷ 12 Side Backing Angle = 24.25713°
D = 22.5° Side Deck Angle = 54.62219°
90° - P2a = Angle between 22.5° Ridge and Cricket Valley Trough = 56.73398°
C5a = 22.5° Side Backing Angle = 12.80070°
Projecting the 12 ÷ 12 Slope through the Deck Angle
arctan (tan 45.00000° sin 54.62219°) = 39.19215°
Angle on Cricket Valley Board = 39.19215° - 18.66136° = 20.53079°
Angle on Common (Jack Rafter) Boards = 45.00000° - 22.50000° = 22.50000°
Calculating with respect to the Cricket Valley Roof Plane
The Dihedral Angle between the side (plumb) face of the Cricket Valley and the Cricket Valley Roof Plane
is 90° + 12.80070° = 102.80070°
LAG Program Entries :
(Set Select Entry Mode to Angular)
Main Pitch : 22.5°
Adjacent Pitch : 102.80070°°
Total Deck Angle : 56.73398°
LAG Program Returns :
90° - P2m = 63.33865°
This answer makes sense; calculating directly from the 12 ÷ 12 slope :
arctan (tan 54.62219° ÷ cos 45.00000°) = 63.33865°
This is the angle on the Level "Sleeper" (on the 12 ÷ 12 slope).
90° - P2a = 159.46921°
NOTE : Adjacent Deck Angle D exceeds 90° : 175.25682°
We require the Supplementary Angle : 180° - 159.46921° = 20.53079°
The angle on the stick (Cricket Valley Rafter), in agreement with the previous result.
Saw Miter Angle = 90° - 20.53079° = 69.46921°
C5m = 10.52909°
C5a = 76.36255°
Blade Bevel Angle = 90° - 76.36255° + 10.52909° = 24.16654°.
This result also makes sense.
The Dihedral Angle between the side face of a Hip-Valley Rafter and the Roof Plane is the complement of the Backing Angle.
Calculating the Backing Angle (and hence, the required Saw Blade Bevel at this intercept) with respect to the initial 12 ÷ 12 slope :
arctan (sin 39.19215° ÷ tan 54.62219°) = 24.16654°.
Oblique "Sleeper" on the 7 ÷ 12 Slope
This piece lies on the intercept of the 7 ÷ 12 Roof Plane and the 22.10606° pitched Cricket Valley Roof Plane;
therefore it follows the 7 ÷ 12 Cricket Valley Trough Line.
The Dihedral Angle between the 7 ÷ 12 Slope and the Cricket Valley Roof Plane
= C5m + C5a = 24.25713° + 12.80070° = 37.05783°
LAG Program Entries :
(Set Select Entry Mode to Angular)
Main Pitch : 22.5°
Adjacent Pitch : 37.05783°
Total Deck Angle : 56.73398°
LAG Program Returns :
90° - P2m = 39.52120°
The angle on the Level "Sleeper" (on the 12 ÷ 12 slope) as it meets the Valley.
90° - P2a = 23.83544°
The angle on the Oblique "Sleeper" (on the 7 ÷ 12 slope) as meets the Valley.
C5m = 17.71971°
C5a = 34.63408°
Note the Sum of the Backing Angles:
C5m + C5a = 17.71971° + 34.63408° = 52.35379°.
Calculating with respect a Level Plane:
LAG Program Entries :
(Set Select Entry Mode to Angular)
Main Pitch : 30.25644°
Adjacent Pitch : 45.00000°
Total Deck Angle : 90°
LAG Program Returns :
Valley Pitch Angle = R1 = 26.74221°
90° - P2m = Angle between 7 ÷ 12 Ridge and Valley Trough = 63.25779°
Angle between 7 ÷ 12 Ridge and Cricket Valley Trough = 39.42235°
Angle on the Oblique "Sleeper" on the 7 ÷ 12 Roof Plane = 23.83544°
39.42235° + 23.83544° = 63.25779°
90° - P2a = Angle between 12 ÷ 12 Ridge and Valley Trough = 39.52120°
In agreement with the calculation from the Cricket Valley Roof Plane.
C5m = 14.70759°
C5a = 37.64621°
Sum of the Backing Angles:
C5m + C5a = 14.70759° + 37.64621° = 52.35380°.
The returned values of the Sum of the Backing Angles agree to four decimal places regardless
of the value chosen for the Cricket Valley Roof Plane Pitch Angle
or the Reference Plane from which the calculation is executed.
Cricket Valley Rafter meets the 7 ÷ 12 Roof Plane
The angle on the Cricket Valley Roof Plane will substitute for the Total Deck Angle.
The value of this angle is 180° - 2 × 56.73398° = 66.53204°.
LAG Program Entries :
(Set Select Entry Mode to Angular)
Main Pitch : 37.05783°
Adjacent Pitch : 102.80070°
Total Deck Angle : 66.53204°
LAG Program Returns :
90° - P2m = 78.84468°
The angle where the "sleeper" on the 7 ÷ 12 slope meets the Cricket Valley.
C5m = 8.31188°
90° - P2a = 142.67728°
NOTE : Adjacent Deck Angle D exceeds 90° : 170.41196°
We require the Supplementary Angle : 180° - 142.67728° = 37.32272°.
This is the angle on the Cricket Valley where it meets the 7 ÷ 12 slope.
Saw Miter Angle = 90° - 37.32272° = 52.67728°.
C5a = 74.05475°
Blade Bevel Angle = 90° - 74.05475° + 8.31188° = 24.25713°
Checking results with respect a Level Plane:
Producing the Plumb Plane of the Cricket Valley (i.e. the side face) to the Deck (or Level Plane)
creates a line at the intercept of the two planes in question.
The 7 ÷ 12 Ridge intercepts this line at an angle of 90° - 54.62219° = 35.37781°.
LAG Program Entries :
(Set Select Entry Mode to Angular)
Main Pitch : 30.25644°
Adjacent Pitch : 90.00000°
Total Deck Angle : 35.37781°
LAG Program Returns :
C5m = 24.25713 °
This result is in accord with the calculation with respect to the Cricket Valley Roof Plane.
