Pyramid Hip Roof Rafters -- Equal & Unequal Pitched Roofs

Rafter Tools Pyramid Hip Roof Framing

Pyramid Hip Roof Framing Input variables
1. Main Pitch Angle
2. Adjacent Pitch Angle
3. Input Length
4. Overhang Run
5. Jack Rafter Spacing
6. Hip Rafter Width
7. Hip Rafter Depth
8. Rafter Width
9. Rafter Depth
10. Seat cut Length

Calculated Results for the Pyramid Hip Roof Rafters. -- Equal & Unequal Pitched Roofs

• Main Plan Angle
• Adjacent Plan Angle
• Main Roof Slope Angle
• Main Roof Pitch
• Adjacent Roof Slope Angle
• Adjacent Roof Pitch
• Hip Rafter Angle
• Hip Rafter Pitch
• Main Hip Rafter Backing Angle
• Adjacent Hip Rafter Backing Angle
• Main Hip Rafter Sidecut Angle
• Adjacent Hip Rafter Sidecut Angle
• Main Pitch Jack Rafter Side Cut Angle.
• Main Pitch Jack Rafter Length Difference @ jack rafter spacing
• Main Pitch Jack Rafter Saw Blade Bevel Angle
• Main Pitch First Jack Rafter Length To Long point of miter cut.
• Adjacent Pitch Jack Rafter Side Cut Angle.
• Adjacent Pitch Jack Rafter Length Difference @ jack rafter spacing
• Adjacent Pitch Jack Rafter Saw Blade Bevel Angle
• Adjacent Pitch First Jack Rafter Length To Long point of miter cut.
• Hip Rafter Run
• Hip Rafter Length
• Hip Rafter Overhang Length.
• Main King Rafter Run
• Main Theoretical King Rafter Length
• Main True King Rafter Length
• Main Rafter Overhang Length
• Adjacent King Rafter Run
• Adjacent Theoretical King Rafter Length
• Adjacent True King Rafter Length
• Adjacent Rafter Overhang Length
• Theoretical Rise
• Rise To Top Of Ridge Peak
• HAP -- Height Above Plate (Heel Height)
• Main Rafter Plumb Cut Height
• Adjacent Rafter Plumb Cut Height
• Hip Rafter Plumb Cut Height
• Hip Rafter Backing Depth Mark.
• Hip Rafter Side Cut Dimension For Side Cut Angles Foot.
• Main Hip Rafter Offset Dimension Perpendicular To Hip Run Line.
• Adjacent Hip Rafter Offset Dimension Perpendicular To Hip Run Line.
• Hip Rafter Backing Offset Main along eave line.
• Hip Rafter Backing Offset Adjacent along eave line.

Math to calculate the shorting of the king rafter run for equal pitched pyramid roofs.

King Rafter Run Shorting At Peak
Plan Angle = 45.00°
King Rafter Run Shorting = (hip rafter width/2) ÷ cos(Plan Angle)
King Rafter Run Shorting = (3.5/2) / cos(45.00°)
King Rafter Run Shorting = 2.4749

This math works for 90° Eave Angles Only, With Equal Pitched Roofs Only

Plan View of Hip Rafter offset for Equal Pitched Pyramid Roof Rafters.

Math to calculate the shorting of the king rafter run for unequal pitched pyramid roofs.

Unequal pitched Pyramid Roof with 4:12 & 8:12 pitched roofs.

King Rafter Run Shorting At Peak for 4:12 Pitch
• Plan Angle =63.43495°
• Hip Offset Along Eave Line = hip rafter width * tan(Plan Angle)
• Hip Offset Along Eave Line =3.5 * sin(63.43495) = 3.130495
• King Rafter Run Shorting = Hip Offset Along Eave Line * tan(Plan Angle)
• King Rafter Run Shorting = 3.130495 * tan(63.43495) = 6.2610

King Rafter Run Shorting At Peak for 8:12 Pitch
• Plan Angle = 26.56505°
• Hip Offset Along Eave Line = hip rafter width * tan(Plan Angle)
• Hip Offset Along Eave Line =3.5 * sin(26.56505) = 1.565248
• King Rafter Run Shorting = Hip Offset Along Eave Line * tan(Plan Angle)
• King Rafter Run Shorting = 1.565248 * tan(26.56505) = 0.782624

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.

Math to calculate the Hip Rafter Sidecut Mark Dimension For Sidecut Angles Foot

Hip Rafter Offset Mark Dimension Perpendicular to Hip Rafter Run Line ÷ tan( plan angle ) ÷cos(hip rafter angle)

Example: 

Hip Rafter Offset Mark Dimension Perpendicular to Hip Rafter Run Line = 2.4231

Plan Angle =  56.30993°

Hip Rafter Angle = 29.01714°

(2.4231 ÷ tan(56.30993°) ) ÷ cos(29.01714°) = 1.847281"

Hip Rafter Sidecut Mark Dimension For Sidecut Angles Foot = 1.847281"


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.

In the Rafter Tools calculator click on the email button to send the Pyramid Hip Rafter Calculations to yourself. The email will look something like this.

Subject: Rafter Tools -- Pyramid Hip Rafter Calculations
Date: Sunday, January 06, 2013 6:03:10 AM
Main Plan Angle
56.30993°
Adjacent Plan Angle
33.69007°
Main Roof Slope Angle
33.69007°
Main Pitch
8 in\12in
Adjacent Roof Slope Angle
45.00000°
Adjacent Pitch
12 in\12in
Hip Rafter Angle
29.01714°
Hip Rafter Pitch
8 in\14 7/16in
Main Hip Rafter Backing Angle
17.92021°
Adjacent Hip Rafter Backing Angle
36.03989°
Main Hip Rafter Side Cut Angle
30.24146°
Adjacent Hip Rafter Side Cut Angle
52.67946°
Main Jack Rafter Side Cut Angle
29.01714°
Main Pitch JackRafter Difference @ 12 ''in
Saw Blade Bevel Angle - 56.30993°
21 5/8''in
Main Pitch First Jack Rafter Length To Long Point
17 3/4''in
Adjacent Jack Rafter Side Cut Angle
46.68614°
Adjacent Pitch JackRafter Difference @ 12 ''in
Saw Blade Bevel Angle - 33.69007°
11 5/16''in
Adjacent Pitch First Jack Rafter Length To Long Point
10 3/16''in
Main King Rafter Run
144 ''in
Main Theoretical King Rafter Length
173 1/16''in
Main True King Rafter Length
167 13/16''in
Main Rafter Overhang Length
28 7/8''in
Hip Rafter Run
173 1/16''in
Hip Rafter Length
197 15/16''in
Hip Rafter Overhang Length
33 ''in
Adjacent King Rafter Run
96 ''in
Adjacent Theoretical King Rafter Length
135 3/4''in
Adjacent True King Rafter Length
133 15/16''in
Adjacent Rafter Overhang Length
33 15/16''in
Rafter Seat Cut Length
3 1/2''in
HAP - Height Above Plate (Heel Height)
9 1/16''in
Main Plumb Height
11 7/16''in
Adjacent Plumb Height
13 7/16''in
Hip Rafter Plumb Height
13 1/8''in
Rise
96 ''in
Rise To Rafter Peak
105 1/16''in
Hip Rafter Layout
Hip Rafter Backing Depth Mark
13/16''in
Hip Rafter Sidecut Dimension For Sidecut Angles Foot
1 7/8''in
Main Hip Rafter Side Cut Angle Dimension Peak
4 3/16''in
Adjacent Hip Rafter Side Cut Angle Dimension Peak
13/16''in
Main Hip Rafter Offset Dimension Perpendicular To Hip Run Line
2 7/16''in
Adjacent Hip Rafter Offset Dimension Perpendicular To Hip Run Line
1 1/16''in
Main Hip Rafter Offset Dimension Along Eave Line
2 15/16''in
Adjacent Hip Rafter Offset Dimension Along Eave Line
1 15/16''in