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

Irregular Pitches, angle in plan between Hip Roof Eaves > 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:

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

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


Joe Bartok
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