This math works for 90° Eave Angles Only
Steps to layout a bastard hip rafter for a pyramid roof.
Unequal pitched Pyramid Roof Example with 8:12 & 12:12 pitched roofs.
Roof Eave Angle = 90.00°
8:12 Roof Slope Angle = 33.69007°
12:12 Roof Slope Angle = 45.00°
Hip Rafter Slope Angle = 29.01714°
8:12 Plan Angle = 56.30993°
12:12 Plan Angle = 33.69007°
Step 1:
First layout the plumb marks on the hip rafter for the hip rafter overhang length and the length of the hip rafter.
Step 2:
This next drawing shows the location of the Main Hip Rafter Offset Mark Dimension Perpendicular
to Hip Rafter Run Line and the location of the Hip Rafter Backing Depth Mark on the un-backed hip rafter. Use these two dimensions to start the layout of the hip rafter.
The hip rafter will not be backed out until all of the layout marks are drawn on the hip rafter material. You need the hip rafter backing depth line drawn on the side of the hip rafter to
establish the hip rafter seat cut location/height.
Step 3:
This next drawing shows the location of the
Hip Rafter Sidecut Mark Dimension For Sidecut Angles Foot and the location of the
Hip Rafter Sidecut Angles at the foot of the hip rafter.
Mark off the
Hip Rafter Sidecut Mark Dimension For Sidecut Angles Foot on the top of the hip rafter.
Then draw the lines back to the center line of the hip rafter where it intersects the plumb line mark.
The hip rafter side cut angles at the foot of the hip rafter are the opposite of the hip rafter side cut angles at the peak of the hip rafter.
Step 4:
This next drawing shows the hip rafter plumb cut and the location of the
Hip Rafters Plumb Backing Angles on the end of the hip rafter in relationship to the hip rafters
backing depth lines on the sides of the hip rafter.
Step 5:
Use the
Main Hip Rafter Offset Mark Dimension Perpendicular
to Hip Rafter Run Line and the
Hip Rafter Sidecut Mark Dimension For Sidecut Angles Foot to draw the side cut angles at the hip rafter's
seat cut location. Draw a plumb line at the hip rafters side cut angles mark. Mark off the common rafters HAP along the plumb line from the hip rafters backing depth line.
Then draw a line that is perpendicular (90°) to the plumb line for the level cut of the seat cut.
Set your saw blade bevel angle to the 33.69007°, to cut the hip rafter tail plumb cut on the 8:12 roof pitch side of the hip rafter. The miter angle of 29.01714° and saw blade bevel angle of 33.69007° produces the Compound
Bevel Angle of 37.32054°. 90.00° - 37.32054 = 52.67946°. Which is the hip rafter side cut angle at the foot of the hip rafter on the 8:12 roof pitch side of the roof.
Set your saw blade bevel angle to the 56.30993°, to cut the hip rafter tail plumb cut on the 12:12 roof pitch side of the hip rafter. The miter angle of 29.01714° and saw blade bevel angle of 56.30993° produces the Compound
Bevel Angle of 59.75854°. 90.00° - 59.75854 = 30.24146°. Which is the hip rafter side cut angle at the foot of the hip rafter on the 12:12 roof pitch side of the roof.
Step 6:
Use the
Main Hip Rafter Side Cut Mark Dimension Peak and the
Adjacent Hip Rafter Side Cut Mark Dimension Peak to draw the hip rafter side cut angles at the hip rafter's
peak.
Math to calculate the
Hip Rafter Side Cut Mark Dimension Peak .
Main Hip Rafter Side Cut Mark Dimension Peak =
Main Hip Rafter Offset Mark Dimension Perpendicular to Hip Rafter Run Line ÷ tan(Main Hip Rafter Side Cut Angle)
Main Hip Rafter Side Cut Mark Dimension Peak =
2 7/16" ÷ tan(30.24146°) = 4 3/16"
Adjacent Hip Rafter Side Cut Mark Dimension Peak =
Adjacent Hip Rafter Offset Mark Dimension Perpendicular to Hip Rafter Run Line ÷ tan(Adjacent Hip Rafter Side Cut Angle)
Adjacent Hip Rafter Side Cut Mark Dimension Peak = 1
1/16" ÷ tan(52.67946°) = 13/16"
Step 7:
Finish cutting the hip rafter's side cut angles at the peak/head of the hip rafter.
Set your saw blade bevel angle to the 56.30993°, to cut the hip rafter head plumb cut on the 8:12 roof pitch side of the hip rafter. The miter angle of 29.01714° and saw blade bevel angle of 56.30993° produces the Compound
Bevel Angle of 59.75854°. 90.00° - 59.75894 = 30.24146°. Which is the hip rafter side cut angle at the peak of the hip rafter on the 8:12 roof pitch side.
Set your saw blade bevel angle to the 33.69007°, to cut the hip rafter head plumb cut on the 12:12 roof pitch side of the hip rafter. The miter angle of 29.01714° and saw blade bevel angle of 33.69007° produces the Compound
Bevel Angle of 37.32054°. 90.00° - 37.32054 = 52.67946°. Which is the hip rafter side cut angle at the peak of the hip rafter on the 12:12 roof pitch side of the roof.
Step 8:
Set your saw blade bevel angle at the
Main Hip Rafter Backing Angle and the
Adjacent Hip Rafter Backing Angle to cut the hip rafter edge bevels along the hip rafter backing depth lines. The hip rafter edge bevels produce the jack rafter side cut angles at the peak of the hip rafter and the hip rafter edge bevels produce the roof sheathing angles at the foot of the hip rafter.
This next drawing shows the bastard hips connected at the peak of the pyramid roof.
This next drawing shows the jack rafter side cut angles at the peak of the pyramid roof peak that are a result of the
hip rafters being backed out.
This next drawing shows geometric development of a pyramid roof.
