Irregular Hip Roof Geometry and Trigonometry
Mathematical Proofs of Plan Angle Formulas

Irregular Hip Roof ... Unequal Pitches, any Eave Angle in Plan View (PDF File)
Compass and Straightedge Solution ... Unequal Roof Slopes with Irregular Plan Angle

 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.

Irregular Pitches, angle in plan between Hip Roof Eaves > 90°
Unequal Pitches, angle in plan between Hip Roof Eaves > 90° ... Major Plan Angle = 90°
Unequal Pitches, angle in plan between Hip Roof Eaves > 90° ... both Plan Angles < 90°
Unequal Pitches, angle in plan between Hip Roof Eaves > 90° ... Major Plan Angle > 90°
3D Models and Sketches ... Unequal Pitches, angle in plan between Hip Roof Eaves > 90° ... one Plan Angle > 90°
 Major Pitch = 8/12 Minor Pitch = 10/12 Plan Angle between Hip Roof Eaves = 120° Major Plan Angle Calculation 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 = 2.88675 Major Plan Angle = arctan 2.88675 = 70.89339° Minor Plan Angle Calculation 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 = 1.25000 tan Minor Plan Angle = .86603 ÷ (1.25000 - .50000) = .86603 ÷ .75000 = 1.15470 Minor Plan Angle = arctan 1.15470 = 49.10661°

Irregular Pitches, angle in plan between Hip Roof Eaves < 90°
 Major Pitch = 5/12 Minor Pitch = 7/12 Plan Angle between Hip Roof Eaves = 75° Major Plan Angle Calculation sin Eave Angle = sin 75° = .96593 cos Eave Angle = cos 75° = .25882 Major Pitch = 5 ÷ 12 = .41667 Minor Pitch = 7 ÷ 12 = .58333 Major Pitch ÷ Minor Pitch = .41667 ÷ .58333 = .71429 tan Major Plan Angle = .96593 ÷ (.71429 + .25882) = .96593 ÷ .97310 = .99262 Major Plan Angle = arctan .99262 = 44.78787° Minor Plan Angle Calculation sin Eave Angle = sin 75° = .96593 cos Eave Angle = cos 75° = .25882 Major Pitch = 5 ÷ 12 = .41667 Minor Pitch = 7 ÷ 12 = .58333 Minor Pitch ÷ Major Pitch = .58333 ÷ .41667 = 1.40000 tan Minor Plan Angle = .96593 ÷ (1.40000 + .25882) = .96593 ÷ 1.65882 = .58230 Minor Plan Angle = arctan .58230 = 30.21213°

 If the angle in plan between the Hip Roof Eaves = 90°, the formulas for the Major Plan Angle and Minor Plan Angle reduce to:

Algebraic Solution: Unequal Pitches, angle in plan between Hip Roof Eaves = 90°
Unequal Pitches, angle in plan between Hip Roof Eaves = 90°
 Major Pitch = 9/12 Minor Pitch = 6/12 Plan Angle between Hip Roof Eaves = 90° Major Plan Angle Calculation sin Eave Angle = sin 90° = 1 cos Eave Angle = cos 90° = 0 Major Pitch = 9 ÷ 12 = .75000 Minor Pitch = 6 ÷ 12 = .50000 Major Pitch ÷ Minor Pitch = .75000 ÷ .50000 = 1.50000 tan Major Plan Angle = 1 ÷ (1.5 + 0) = 1 ÷ 1.50000 = .66667 Major Plan Angle = arctan .66667 = 33.69007° Minor Plan Angle Calculation sin Eave Angle = sin 90° = 1 cos Eave Angle = cos 90° = 0 Major Pitch = 9 ÷ 12 = .75000 Minor Pitch = 6 ÷ 12 = .50000 Minor Pitch ÷ Major Pitch = .50000 ÷ .75000 = .66667 tan Minor Plan Angle = 1 ÷ (.66667 + 0) = 1 ÷ .66667 = 1.50000 Minor Plan Angle = arctan 1.50000 = 56.30993°

 If the pitches are equal geometry guarantees that the angle in plan between the Hip Roof Eaves is bisected. Major Plan Angle = Minor Plan Angle and the formulas reduce to:

Equal Pitches, angle in plan between Hip Roof Eaves > 90°
 Major Pitch = 8/12 Minor Pitch = 8/12 Plan Angle between Hip Roof Eaves = 135° Plan Angle Calculation sin Eave Angle = sin 135° = .70711 cos Eave Angle = cos 135° = - .70711 Major Pitch = 8 ÷ 12 = .66667 Minor Pitch = 8 ÷ 12 = .66667 Major Pitch ÷ Minor Pitch = .66667 ÷ .66667 = 1 tan Plan Angle = .70711 ÷ (1 - .70711) = .70711 ÷ .29289 = 2.41421 Plan Angle = arctan 2.41421 = 67.50000°