The Hip rafter are perpendicular to one another.

The sums of the Plan Angles are right angles: DDa + DDb = 90°

dd

_{a}+ ddb = 90°

For a given roof, the sum of the angles on the roof surface between the Hip rafter ridge lines is

fixed:

Σ P2 = arctan (cos SS ÷ tan DDa) + arctan (cos SS ÷ tan DD

_{b})

Backing Angles Formulas:

C5a = arctan (sin R1

_{a}÷ tan DD

_{a}) C5

_{b}= arctan (sin R1

_{b}÷ tan DD

_{b})

The Backing Angles are also related by the formulas: C5

_{a}= arctan (cos Σ P2 ÷ tan C5)

C5b = arctan (cos Σ P2 ÷ tan C5

_{a})

The angles at the Hip rafter feet – R4B, R5B and A5B – are defined by the formulas for a standard

Hip-Valley Roof.

For a given roof of this type: R5B

_{a}= R5B

_{b}

Definitions of Angles, Formulas

and an Example Calculation

SS = Common Rafter Slope Angle

= 31.71748°

DD = Plan Angle at Hip Rafter Foot

= 31.71748°

dd = Plan Angle at Hip Rafter Peak

= 40°

α = C5 = Hip Rafter Backing Angle

= 26.56505°

β = 90° – R1

= 90° – Hip Rafter Slope Angle

= 72°

μ = arctan (sin β tan dd) = arctan (cos R1 tan dd)

arctan (sin 72° tan 40°) = 38.59096°

r1 = Section Plane Slope Angle

= arctan (tan SS sin (DD + dd))

= arctan (tan 31.71748° sin (31.71748° + 40°))

= 30.40596°

Blade Bevel for XDIH

= Blade Bevel along MIT Line

= Section Plane Backing Angle

= arctan (sin R1 ÷ tan (DD + dd))

= arctan (sin 30.40596° ÷ tan (31.71748° + 40°))

= 9.49278°

Projected Right Angles

= 90° ± arctan (cos dd ÷ tan ß)

= 90° ± arctan (cos dd tan R1)

= 90° + arctan (cos 40° ÷ tan 72°) = 103.97707°

= 90° – arctan (cos 40° tan 18°) = 76.02293°

ρ = r1 – arctan (cos dd tan R1)

= 30.40596° – arctan (cos 40° tan 18°) = 16.42889°

Blade Bevel for YDIH

= Blade Bevel along ß Line

= 90° – dd = 50°

Blade Bevel for ZDIH

= Blade Bevel along μ Line

= arctan (cos Projected Right Angle tan dd)

= arctan (cos 76.02293° tan 40°) = 11.45699°

= arccos (sin ß ÷ sin Projected Right Angle)

= arccos (sin 72° ÷ sin 76.02293°)

= 11.45699°

hα = hρ = .286364

hα = √ (sin MIT)

^{2}+ (cos MIT tan μ)

^{2}– 2 sin MIT cos MIT tan μ cos α

hρ = √ 1 + (cos MIT ÷ cos μ)

^{2}– 2 cos MIT cos ρ ÷ cos μ

Non-Rectangular Sections Calculator Examples

Plan Angles at Peak arbitrarily set at 40° and 50°

18° Hip Entries:

Plan Angle at Peak = 40°

μ = arctan (cos 18° tan 40°) = 38.59096°

ß = 72° (90° – R1)

α = 26.56505° (C5)

Returns (rounded off):

MIT = 38.30262° ... Angle on Roof Surface

BEV = 59.59404° ... 90°– Section Plane Angle

ρ = 16.42889° ... Section Plane Angle – (90° – 76.02293°) Projected Right Angle = 103.97707°

Supplementary Angle = 76.02293°

Blade Angle along MIT Line = 9.49278°

Blade Angle along ß Line = 50.00000°

Blade Angle along μ Line = 11.45699°

27.7323° Hip Entries: Plan Angle at Peak = 50°

μ = arctan (cos 27.7323° tan 50°) = 46.52926°

ß = 62.2677° (90° – R1)

α = 16.04506° (C5)

Returns (rounded off):

MIT = 43.42969° (Angle on Roof Surface)

BEV = 59.59404° ... 90°– Section Plane Angle)

ρ = 11.73413° ... Section Plane Angle – (90° – 71.32817°) Projected Right Angle = 108.67183°

Supplementary Angle = 71.32817°

Blade Angle along MIT Line = 9.49278°

Blade Angle along ß Line = 40.00000°

Blade Angle along μ Line = 20.88368°

90° Hip Run intersection bisected (45°)

Equal Hip Rafter Widths (solution by Sim Ayers)

18° Hip Entries:

Plan Angle at Peak = 45°

μ = arctan (cos 18° tan 45°) = 43.563° (R4P)

ß = 72° (90° – R1)

α = 26.56505° (C5)

Returns (rounded off):

MIT = 42.64537° ... Angle on Roof Surface

BEV = 58.97305° ... 90°– Section Plane Angle

ρ = 18.08763° ... Section Plane Angle – (90° – 77.06068°) Projected Right Angle = 102.93932°

Supplementary Angle = 77.06068°

Blade Angle along MIT Line = 6.93760°

Blade Angle along ß Line = 45.00000°

Blade Angle along μ Line = 12.62142°

27.7323° Hip Entries: Plan Angle at Peak = 45°

μ = arctan (cos 27.7323° tan 45°) = 41.51305° (R4P)

ß = 62.2677° (90° – R1)

α = 16.04506° (C5)

Returns (rounded off):

MIT = 39.08693° ... Angle on Roof Surface

BEV = 58.97305° ... 90°– Section Plane Angle

ρ = 10.63443° ... Section Plane Angle – (90° – 69.60748°) Projected Right Angle = 110.39252°

Supplementary Angle = 69.60748°

Blade Angle along MIT Line = 6.93759°

Blade Angle along ß Line = 45.00000°

Blade Angle along μ Line = 19.21087°

Standard Hip-Valley Angles

18° Hip Entries:

Plan Angle at Peak = 90° – DD = 58.28252°

μ = arctan (cos 18° tan 58.28252°) = 56.98260° (R4P)

ß = 72° (90° – R1)

α = 26.56505° (C5)

Returns (rounded off):

MIT = 54° (P2 ... Angle on Roof Surface)

BEV = 58.28253° (90°– SS ... Section Plane Angle)

ρ = 22.02375° (SS – R5P)

Projected Right Angle = 99.69372° (90° + R5P)

Supplementary Angle = 80.30628° (90° – R5P) Blade

Angle along MIT Line = 0.00000°

Blade Angle along ß Line = 31.71748° (90° – DD ... Blade Bevel along 90° – R1)

Blade Angle along μ Line = 15.24016° (A5P ... Blade Bevel along R4P)

27.7323° Hip Entries:

Plan Angle at Peak = Peak = 90° – DD = 31.71748°

μ = arctan (cos 27.7323° tan 31.71748°) = 28.68049° (R4P)

ß = 62.2677° (90° – R1)

α = 16.04506° (C5)

Returns (rounded off):

MIT = 27.73230° (P2 ... Angle on Roof Surface)

BEV = 58.28253° (90°– SS ... Section Plane Angle)

ρ = 7.62264° (SS – R5P)

Projected Right Angle = 114.09484° (90° + R5P)

Supplementary Angle = 65.90516° (90° – R5P) Blade

Angle along MIT Line = 0.00000°

Blade Angle along ß Line = 58.28252° (90° – DD ... Blade Bevel along 90° – R1)

Blade Angle along μ Line = 14.16082° (A5P ... Blade Bevel along R4P)

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