Solution of Irregular Hip Roof
Plan Angle Formula

 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.

 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.

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:

 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.

 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°

 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)