Solution of Irregular Hip Roof
Plan Angle Formula
Section removed from Irregular Hip Roof
Irregular Hip Roof Model: Trigonometric Scale
Irregular Hip Roof Model in Plan View
An angle inscribed in a semi-circle is a right angle.
The quadrilateral in plan, bounded by the eaves, Major Side Common Run and Minor Side Common Run, may be circumscribed.

A quadrilateral inscribed in a circle has supplementary opposite angles.
Two opposite angles are right angles, hence the remaining two angles are supplementary. The angle shown as the Eave Angle to the upper right is correct.

Angles inscribed in a circle which subtend the same arc are equal.
The angle shown as the Major Plan Angle to the upper left is correct.

Irregular Hip Roof: Major Plan Angle Formula
Since the tangents of the Common Pitch Angles are their Rise/Run ratios the Plan Angle formula may be written in terms of the Common pitches.

Irregular Hip Roof: Major Plan Angle Formula in terms of Common Pitches

Resolving the Dimensions
Major Plan Angle Calculation
Major Pitch = 8/12
Minor Pitch = 10/12
Plan Angle between Hip Roof Eaves = 120°

sin Eave Angle = sin 120° = .86603
cos Eave Angle = cos 120° = - .50000
Major Pitch = 8 ÷ 12 = .66667
Minor Pitch = 10 ÷ 12 = .83333
Major Pitch ÷ Minor Pitch = .66667 ÷ .83333 = .80000
tan Major Plan Angle = .86603 ÷ (.80000 - .50000)
= .86603 ÷ .30000 = 2.88677

2.88677 represents the Rise per Unit Run of the Major Plan Angle.
The Major Plan Angle Rise is also the Major Common Run (refer to the Plan View above).
If the Major Plan Angle Rise = Major Common Run = 12
Then the Major Plan Angle Run along the eave line
= Major Plan Angle Rise ÷ 2.88677
= 12 ÷ 2.88677 = 4.15690
This is sufficient information to calculate the Hip Run and the balance of the roof dimensions.

The Plan Angles also govern the Saw Blade Bevels for the cheek cuts on the Hip rafter and Jack rafters.
Since tan Major Plan Angle = 2.88677
Major Plan Angle = arctan 2.88677 = 70.89351°
Minor Plan Angle = Eave Angle - Major Plan Angle
= 120° - 70.89351° = 49.10649°

The Rise per Unit Run of the Minor Plan Angle
= tan Minor Plan Angle = tan 49.10649° = 1.15470
The ratio may also be determined directly from the formula:

Irregular Hip Roof: Minor Plan Angle Formula in terms of Common Pitches
Minor Plan Angle Calculation
Major Pitch = 8/12
Minor Pitch = 10/12
Plan Angle between Hip Roof Eaves = 120°

sin Eave Angle = sin 120° = .86603
cos Eave Angle = cos 120° = - .50000
Major Pitch = 8 ÷ 12 = .66667
Minor Pitch = 10 ÷ 12 = .83333
Minor Pitch ÷ Major Pitch = .83333 ÷ .66667 = 1.24999
tan Minor Plan Angle = .86603 ÷ (1.24999 - .50000)
= .86603 ÷ .74999 = 1.15472

Solving the Plan Dimensions
Known Values
Major Common Run = 12
Major Common Eave = 4.15690
Rise = 8

Minor Common Run = 8 × 12 ÷ 10 = 9.60000
Hip Rafter Run ² = Major Common Eave ² + Major Common Run ²
= 4.15690 ² + 12 ² = 161.27982
Hip Rafter Run = 12.69960
Minor Common Eave ² = Hip Rafter Run ² - Minor Common Run ²
= 161.27982 - 9.60000 ² = 69.11982
Minor Common Eave = 8.31383
Checking the Minor Common Eave length:
Since the Minor Plan Angle Rise per Unit Run = 1.15470
Minor Common Eave = Minor Common Run ÷ 1.15470
= 9.60000 ÷ 1.15470 = 8.31385

Solving the Rafter Lengths
Major Common Length ² = Major Common Run ² + Rise ²
= 12 ² + 8 ² = 208.00000
Major Common Length = 14.42221
Minor Common Length ² = Minor Common Run ² + Rise ²
= 9.60000 ² + 8 ² = 156.16000
Minor Common Length = 12.49640
Hip Rafter Length ² = Hip Rafter Run ² + Rise ²
= 161.27982 + 8 ² = 225.27982
Hip Rafter Length = 15.00932

NOTE: The dimensions and angles in the sketch below were solved using the online calculator.
Values may differ slightly from those determined by the Pythagorean Theorem and proportion.

Irregular Hip Roof Dimensions
Irregular Hip Roof Sheathing Layout
Jack Rafter Difference in Length
Given: Jack Rafter spacing on centers along eave line = 16"

Major (8/12) Side
16" × 14.42221 ÷ 4.15692 = 55.51114" (approx. 55 1/2")
Minor (10/12) Side
16" × 12.49640 ÷ 8.31384 = 24.04934" (approx. 24 1/16")

Hip Shift and Hip Drop
Major Plan Angle Unit Rise/Run = 2.88677
Minor Plan Angle Unit Rise/Run = 1.15470
Major Side Shift Unit Ratio = 2.88677 ÷ (2.88677 + 1.15470) = .71429
Minor Side Shift Unit Ratio = 1.15470 ÷ (2.88677 + 1.15470) = .28571
Shift to align ridge with line through Framing Points
= .5 × (.71429 - .28571) = .21429 (to 8/12 Side)
Unit Hip Drop = 8 ÷ [12.69961 × (2.88677 + 1.15470)] = .15587

The Unit Ratios are multiplied by the stock width. For 1½" wide stock:
Hip Shift (to 8/12 Side) = 1.5" × .21429 = .32143" (approx. 5/16")
Hip Drop = 1.5" × .15587 = .23381" (approx. 1/4")

Backing the Hip Rafter
Major (8/12) Side
Framing Square
12 × [8 ÷ (2.88677 × 15.00933)] = 2.21563 (approx. 2 3/16 over 12)
Backing Angle = arctan [8 ÷ (2.88677 × 15.00933)] = 10.46105°
Minor (10/12) Side
Framing Square
12 × [8 ÷ (1.15470 × 15.00933)] = 5.53912 (approx. 5 9/16 over 12)
Backing Angle = arctan [8 ÷ (1.15470 × 15.00933)] = 24.77773°

Irregular Hip Shift and Hip Drop
Over 12 ratios for Hip Rafter Side Cuts
Major (8/12) Side
Hip Rafter Foot
12 × 12.69961 × 2.88677 ÷ 15.00933 = 29.31045 (approx. 29 5/16 over 12)
Hip Rafter Peak
12 × 12.69961 ÷ (2.88677 × 15.00933) = 3.51721 (approx. 3 1/2 over 12)
Minor (10/12) Side
Hip Rafter Foot
12 × 12.69961 × 1.15470 ÷ 15.00933 = 11.72410 (approx. 11 3/4 over 12)
Hip Rafter Peak
12 × 12.69961 ÷ (1.15470 × 15.00933) = 8.79308 (approx. 8 13/16 over 12)


Joe Bartok
1