Purlins following a Sloped Ridge Line
intercept plane of Main Roof

Eyebrow Dormer with Sloped Ridge: Substitute Angles for Main Roof Angle
Supplementary Reading:
Dormer with Sloped Ridge Line
Purlins following an Elliptic Arc


Purlins following Sloped Ridge Line: Plan View
Purlins following Sloped Ridge Line: Section View
Plan Angles
Angles on the Planes of the Roof
Angles in Section View
Resolving the Principal Framing Angles

NOTE : The following calculations may be validated independently on any scientific calculator with trig functions or a Construction Master calculator.

GIVEN :
All Common Rafter Pitches = 5 ÷ 12
Dormer Ridge Slope = 2 1/2 ÷ 12 as viewed in elevation
Main Ridge and Dormer Ridge intercept = 90° as viewed in plan

Solution of Plan Angles or Deck Angles
The Dormer Ridge is equivalent to a Hip Rafter Ridge or Valley Rafter Trough
Hip-Valley Pitch = Common Pitch × sin Deck Angle
Therefore sin Deck Angle = Hip-Valley Pitch ÷ Common Pitch
and Deck Angle = arcsin (Hip-Valley Pitch ÷ Common Pitch)
= arcsin (2.5 ÷ 5) = arcsin (.5) = 30°
The Total Deck Angle between the Dormer Eaves = 2 × 30° = 60°
The Total Deck Angle between the Dormer Eave and the Main Eave = 240°

Lag Calculator Entries :
5 ÷ 12 , 5 ÷ 12 , 60°

Click on or button (equal Common Pitches; Main Side = Adjacent Side)
SS = Common Pitch Angle = 22.61986°
DD = Deck Angle or Plan Angle = 30.00000°
R1 = Hip-Valley Pitch Angle = 11.76829° (= 2 1/2 ÷ 12 , as required)
90° - P2 = Angle on Roof Plane between Dormer Eave and Dormer Ridge
= Sheathing Angle = 32.02445°
C5 = Backing Angle for Dormer Ridge Line = 19.45623°

Lag Calculator Entries :
5 ÷ 12 , 5 ÷ 12 , 240°

Click on or button (equal Common Pitches; Main Side = Adjacent Side)
SS = Common Pitch Angle = 22.61986°
DD = Deck Angle or Plan Angle = 120.00000°
R1 = Hip-Valley Pitch Angle = 19.84167° (= 4 11/32 ÷ 12)
90° - P2 = Angle on Roof Plane between Valley Trough and Main Ridge
= Sheathing Angle = 61.94512°
C5 = Backing Angle for Valley Trough Line = 11.08749°

Angles on the Triangle of the Dormer Roof Plane :
Angle on Roof Plane between Valley Trough and Main Eave
= 180° - 61.94512° = 118.05488°
Angle on Roof Plane between Valley Trough and Dormer Ridge
= 180° - (118.05488° + 32.02445°) = Sheathing Angle = 29.92067°

Resolving the Compound Angles on the Dormer Purlins
Purlins intercept Main Roof Calculator

Purlins intercept slope of Main Roof
Calculator Entries
Select Pitch Angle mode
Main Roof Rise = 10.85158° (Section View, angle of intercept with Main Roof)
Dormer Rise = 19.45623° (the Backing Angle along the Dormer Ridge Line, therefore the Dormer Pitch Angle as viewed in section)

Calculator Returns
Face set in Dormer Roof
Saw Blade Bevel = 67.82502° (complement of 2 × Backing Angle
= Sleeper Angle = 2 × 11.08749° = 22.17498°
= Dihedral Angle between the planes of the Dormer Roof and the 5/12 Main Roof)
Angle on the Stick = 29.92068° (angle between Dormer Ridge and Valley Trough)
Sheathing Angles on Main Roof Plane
Sheathing Angle = 61.94511° (angle between Main Ridge and Valley Trough)

Purlins intercept level plane of Deck
Calculator Entries
Select Pitch Angle mode
Main Roof Rise = 11.76829° (Section View, angle of intercept with Deck)
Dormer Rise = 19.45623° (the Backing Angle along the Dormer Ridge Line, therefore the Dormer Pitch Angle as viewed in section)

Calculator Returns
Face set in Dormer Roof
Saw Blade Bevel = 67.38014° (complement of the specified 5/12 Common Pitch Angle)
Angle on the Stick = 32.02445° (angle measured on the plane of the roof between Dormer Ridge and Deck)
Sheathing Angles on Main Roof Plane
Sheathing Angle = 59.99999° (angle measured on the plane of the Deck, complement of the specified 30° Dormer Deck Angle)

Joe Bartok
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