Trirectangular Tetrahedron Study Joe Bartok

The Rules of the Trirectangular Tetrahedron Game
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)