Calculating with respect to a level plane through the Ridges
Common Pitch Angle = arctan (7 ÷ 12) = 30.25644°
Valley Pitch Angle = arctan (4 ÷ 12) = 18.43495°
7 ÷ 12 Side Deck Angle = arcsin (4 ÷ 7) = 34.84990°
Cricket Deck Angle = 90° - 34.84990° = 55.15010°
Cricket Common Pitch Angle = arctan (tan 18.43495° ÷ sin 55.15010°) = 22.10606°
Backing Angle = arctan (sin 18.43495° ÷ tan 55.15010°) = 12.41795°
Dihedral Angle = 90° - 12.41795° = 77.58205°
Angle on Cricket Roof Plane between Level "Sleeper" (on 12 ÷ 12 Slope) and Cricket Valley Trough
= arctan (tan 55.15010° ÷ cos 22.10606°) = 57.17297°
Projecting the 12 ÷ 12 Slope through the Deck Angle
arctan (tan 45.00000° sin 55.15010°) = 39.37408°
Angle on Cricket Valley Board = 39.37408° - 18.43495° = 20.93913°
Angle on Common (Jack Rafter) Boards = 45.00000° - 22.10606° = 22.89394°
Calculating with respect to the Cricket Valley Roof Plane
IMAGES: Geometric Model of Cricket Valley
LAG Program Entries :
(Set Select Entry Mode to Angular)
Main Pitch : 22.89394°
Adjacent Pitch : 180° - 77.58205° = 102.41795°
Total Deck Angle : 57.17298°
LAG Program Returns :
90° - P2m = 63.78595°
This answer makes sense; calculating directly from the 12 ÷ 12 slope :
arctan (tan 55.15010° ÷ cos 45.00000°) = 63.78594°
This is the angle on the Level "Sleeper" (on the 12 ÷ 12 slope).
90° - P2a = 159.06087°
We require the Supplementary Angle : 180° - 159.06087° = 20.93913°
The angle on the stick, in agreement with the previous result.
Saw Miter Angle = 90° - 20.93913° = 69.06087°
C5m = 10.56636°
C5a = 76.73406°
NOTE : Adjacent Deck Angle D exceeds 90° : 175.29604°
Blade Bevel Angle = 90° - 76.73406° + 10.56636° = 23.83230°.
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 39.37408° ÷ tan 55.15010°) = 23.83230° ... right on the money!
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 Total Deck Angle between "ridges" on the level plane is 90°.
LAG Program Entries :
(Set Select Entry Mode to Angular)
Main Pitch : 22.10606°
Adjacent Pitch : 30.25644°
Total Deck Angle : 90°
LAG Program Returns :
C5m = 12.41795°
C5a = 24.42506°
The Sum of the Backing Angles is the Dihedral Angle
between the Cricket Valley Roof Plane and the 7 ÷ 12 Roof Plane:
C5m + C5a = 12.41795° + 24.42506° = 36.84301°.
Calculating with respect to the Cricket Valley Roof Plane:
The 57.17298° angle on the Cricket Valley Roof Plane will substitute for the Total Deck Angle.
LAG Program Entries :
(Set Select Entry Mode to Angular)
Main Pitch : 22.89394°
Adjacent Pitch : 36.84301°
Total Deck Angle : 57.17298°
LAG Program Returns :
90° - P2m = 39.52120°
The angle on the Level "Sleeper" (on the 12 ÷ 12 slope) as it meets the Valley.
C5m = 18.04309°
90° - P2a = 24.38477°
The angle on the Oblique "Sleeper" (on the 7 ÷ 12 slope) as meets the Valley.
C5a = 34.31070°
Note the Sum of the Backing Angles:
C5m + C5a = 18.04309° + 34.31070° = 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°
C5m = 14.70759°
C5a = 37.64621°
Sum of the Backing Angles:
C5m + C5a = 14.70759° + 37.64621° = 52.35380°
; both calculations are equal to four decimal places ... close enough!
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 × 57.17298° = 65.65404°.
LAG Program Entries :
(Set Select Entry Mode to Angular)
Main Pitch : 36.84301°
Adjacent Pitch : 180° - 77.58205° = 102.41795°
Total Deck Angle : 65.65404°
LAG Program Returns :
90° - P2m = 77.74603°
The angle where the "sleeper" on the 7 ÷ 12 slope meets the Cricket Valley.
C5m = 9.03599°
90° - P2a = 143.13011°
We require the Supplementary Angle : 180° - 143.13011° = 36.86989° ( 9 ÷ 12 pitch ).
This is the angle on the Cricket Valley where it meets the 7 ÷ 12 slope.
Saw Miter Angle = 90° - 36.86989° = 53.13011°.
C5a = 74.61092°
NOTE : Adjacent Deck Angle D exceeds 90° : 170.83818°
Blade Bevel Angle = 90° - 74.61092° + 9.03599° = 24.42507°
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° - 55.15010° = 34.84990°.
LAG Program Entries :
(Set Select Entry Mode to Angular)
Main Pitch : 30.25644°
Adjacent Pitch : 90.00000°
Total Deck Angle : 34.84990°
LAG Program Returns :
C5m = 24.42506°
This result is in accord with the calculation with respect to the Cricket Valley Roof Plane to four decimal places.
