Hip & Valley Roof Framing Example # 1

This Hip & Valley Roof  Framing  article will cover the different steps in calculating the roof framing lengths and angles for:

Equal Pitched Roof 6:12
Hip Rafter Width = 3.5"
Rafter Width = 1.5"

I'll use use my Android Rafter Tools app and  my Rafter Tools+ for iPhone app to check the calculations I come up with using geometry and trigonometry.

What are the Valley Jack Rafter Lengths for A & B ?

What are the Valley Jack Rafter Slider (Doppelschifter )Lengths for Jack Rafters C, D & E ?

What are the Hip Rafter Lengths for the Bay Window Hip Rafters A & B & the Cripple Hip Rafter ?

What is the Miter & Edge Bevel Angles for the California Valley Rafter Sleeper?

What is the Valley Rafter Slope Angle for the Dog Leg Valley ?

What are the Frieze Block Angles ?

What are the Purlin Miter & Bevel & Lip Cut Angles ?

What are the Hip Rafter Diamond Post Miter & Bevel Angles ?

What are the King Common Rafter Lengths per structural Detail 1?

Using Geometry, draw out the triangles representing each framing member to calculate-dimension  all of the theoretical lengths of the rafters. You could layout the roof full scale with each framing member drawn to it's actual width if the walls are not yet framed.

1. Red Triangles are Common Rafters
2. Blue Triangles are Hips
3. Green Triangles are Jack Rafter Sliders
4. Yellow Triangles are Valley Sleepers
5. Orange Triangles are Cripple Hips

There are a lots of ways of calculating the common rafter lengths and the hip/valley rafter lengths.

1. Using Geometry
2. Radford's Cyclopedia Of Construction
3. Roof Framing by H.H. Siegele
4. Steel Square by Gilbert Townsend
5. Practical Use Of the Steel Square by Fred Hodgson
6. Handbook of Carpentry and Joinery by A.B. Emary
7. American Carpenter And Builder Magazine
8. Carpenter's Framing Square
9. Chappell Master Framing Square
10. Advanced Timber Framing by Steve Chappell
11. When Roofs Collide by Will Beemer
12. Timber Framing Guild Publications 
13. Full Length Roof Framer by A.J. Riechers
14. Roof Framer by Marshall Gross
15. Roof Framer's Bible by Barry Mussell
16. The Rafter Book by David McIntire
17. A Roof Cutter's Secrets by Will Holladay
18. A Scientific Calculator 
19. A Graphing Calculator like the HP 50g
20. Construction Master Calculator
21. Joe Bartok's Online Framing Calculators
22. Greg Tarrant's Online Framing Calculators
23. BuildCalc app for Android & iPhone
24. Rafter Tools app for Android
25. Rafter Tools+ app for iPhone
26. Basiswissen Schiten
27. Grundwissen des Zimmerers
28. La Charpente en Bois by Emery
29. Complex Roof Framing by Billy Dillon
30. Framing a Hip Roof by Tim Uhler
31. Trigonometry

Example using Geometry

Standard Roof Framing Kernel

Tetrahedron Roof Framing Kernel

Complex Roof Framing Kernel based on a Prism, the unbacked shoulder of  the Hip Rafter

Roof Framing Geometry Kernel rafter lengths roof ratio multiplier.
The common rafter length can be calculated by the roof ratio length per 12" of run.  Divide the rafter length by 12" of run to calculate the roof ratio multiplier.

Hip Rafter Ratio Length Multiplier
HRRLM = 18 ÷ 12 = 1.5

Common Rafter Length Multiplier
CRLM = 13.4164 ÷ 12 = 1.11803

King Common Rafter Length TL

[Decimal Inch] 178" x 1.11803 = 199.00934" TL

Hip Rafter Length TL

[Decimal Inch] 178" x 1.5 = 267" TL

Example using trigonometry

TL = Theoretical Length or True Length of Rafter

FL = Framing - Cut Length of Rafter

King Common Rafter Length
6:12 Pitch

Plan Angle = 45°
Rafter Slope Angle 26.56505°

Hip Rafter 6:17 or 6:16.97 or 19.47122°

14'-10" Common Rafter Run
or
14'-9 1/4" Common Rafter Run to the Ridge

King Common Rafter Length TL

Common Rafter Run ÷ cos ( Roof Slope Angle )
[Feet Inches] 14'-10" ÷ cos (26.56505°) = 16'-7" TL

[Decimal Inch] 178" ÷ cos (26.56505°) = 199.01005 TL

King Common Rafter Length To the Ridge FL

Common Rafter Run To the Ridge ÷ cos ( Roof Slope Angle )
[Feet Inches]  14'-9 1/4" ÷ cos (26.56505°) = 16' 6 3/16" FL
[Decimal Inch] 177.25" ÷ cos (26.56505°) = 198.1715" FL

Theoretical Hip Rafter Run 

Common Rafter Run ÷ cos ( Plan Angle )

[Feet Inches] 14'-10" ÷ cos (45°) = 20'-11 3/4" 

[Decimal Inch] 178" ÷ cos (45°) = 251.73001" 

Hip Rafter Run To Ridge

Common Rafter Run To Ridge ÷ cos ( Plan Angle )

[Feet Inches]   14'-9 1/4"  ÷ cos (45°) = 20'-10 11/16" 

[Decimal Inch] 177.25"  ÷ cos (45°) = 250.66935" 

Hip Rafter Length  TL

Hip Rafter Run ÷ cos ( Hip Rafter Slope Angle )
[Feet Inches] 20'-11 3/4" ÷ cos (19.47122°) = 22'-3" TL
[Decimal Inch] 251.73001"   ÷ cos (19.47122°) =  266.99999" TL

Hip Rafter Length To The Ridge FL

Hip Rafter Run  to the Ridge ÷ cos ( Hip Rafter Slope Angle )
[Feet Inches] 20'-10 11/16" ÷ cos (19.47122°) = 22'-1 7/8" FL
[Decimal Inch] 250.66935"  ÷ cos (19.47122°) = 265.87499" FL

I think the easiest way to calculate the lengths of the Common Rafter, Hip Rafter and Jack Rafter Difference is to use the roof sheathing angle.
Roof Sheathing Angle = arctan (tan Plan Angle ÷ cos Common Rafter Slope Angle)
Roof Sheathing Angle = arctan (tan 45° ÷ cos 26.56505°)
48.18968° = arctan (tan 45° ÷ cos 26.56505°)

The tan of 45° = 1, so you can 1÷ cos Common Rafter Slope Angle
or in this example
48.18968° = arctan (1 ÷ cos 26.56505°)

For the Common Rafter Length to the Ridge
[Decimal Inch] 177.25"  x tan (48.18968°) = 198.17148" FL

For the Hip Rafter Length to the Ridge
[Decimal Inch] 177.25"  ÷ cos (48.18968°) = 265.87497" FL

For the Jack Rafter Length Difference
[Decimal Inch] 24"  x tan (48.18968°) =  26.83281"

For the Jack Rafter Spacing Marks on the Hip Rafter
[Decimal Inch] 24"  ÷ cos (48.18968°) = 35.99999"

For the Purlin Mark on the Hip Rafter

Run To Purlin in Plan View  ÷ cos (48.18968°) = Purlin Mark on the Hip Rafter

For the Cripple Hip Rafter Length TL
First calculate the  hip rafter length for the 14'-8" run
[Decimal Inch] 178"  ÷ cos (48.18968°) = 266.99999" TL

Next calculate the  hip rafter length for the 16'-10 1/4" run
[Decimal Inch] 202.25"  ÷ cos (48.18968°) = 303.37496" TL

Cripple Hip Rafter Length TL = 303.37496" - 266.99999"  = 36.37506" TL

or use the common rafter runs to calculate the cripple hip rafter length

202.25 - 178  = 24.25" Run for the cripple hip rafter

24.25 ÷ cos (48.18968°) = 36.37499" TL

deduct the ridge thickness for both ridges, 0.75 + 0.75 = 1.5

24.25" - 1.5" ÷ cos (48.18968°) = 34.12499" FL

The Bay Window pop-out can be framed 5 or 6 different ways, with the hip rafters offset from the corner for an equal overhang, but with exposed frieze blocks it's easier to leave the Bay Window hip rafters centered on the 45° Bay Window pop-out walls. The Bay Window King Common Rafter Pitch is the same as the rest of the roof and has the same overhang run on the Bay Window Front Wall. Use the Span of the Bay Window to calculate the rise of the Bay Window roof. In this example the pitch is 6:12 and the Span is 10'-0", the offset is 24" and the Bay Window Front Wall Length is 72". First calculate the Bay Window plan angles. The eave angle is 135°. 

Bay Window Plan Angles

Front Wall Plan Angle = arctan( 60" ÷ 36" ) = 59.03624°

Side Wall Plan Angle = 135° - 59.03624° = 75.96375°

Hip Rafter Slope Angle A & B

Hip Rafter Slope Angle  = arctan (tan Common Rafter Pitch Angle × sin Plan Angle)

Hip Rafter Slope Angle  = arctan (tan 26.56505° × sin 59.03624°) = 23.20706°

Roof Sheathing Angle = arctan (tan Plan Angle ÷ cos Common Rafter Slope Angle)
Roof Sheathing Angle = arctan (tan 59.03624° ÷ cos 26.56505°)
61.77948° = arctan (tan 59.03624° ÷ cos 26.56505°)

For the Common Rafter Length
[Decimal Inch] 36"  x tan (61.77948°) = 67.08204" TL

For the Hip Rafter Length
[Decimal Inch] 36"  ÷ cos (61.77948°) = 76.13146" TL

For the Hip Rafter Length To The Ridge
[Decimal Inch] 35.25"  ÷ cos (61.77948°) = 74.54537" FL

For the Jack Rafter Length Difference
[Decimal Inch] 24"  x tan (61.77948°) =  44.72135"

For the King Common Rafter Length
(Ridge Width x 0.5) x tan (Plan Angle) = Ridge Deduction in Plan View

(1.5 x 0.5) x tan (59.03624) = 1.25"

[Decimal Inch] 60" - 1.25"  ÷  cos (26.56505°) = 65.68449" FL

For the Side Wall King Common Rafter Slope Angle

Side Wall Plan Angle = 75.96375°

Hip Rafter Slope Angle  = 23.20706°

Side Wall Length = 24" ÷ cos( 45° ) = 33.94113"

Side Wall King Common Rafter Slope Angle = arctan (tan Hip-Valley Pitch Angle ÷ sin Plan Angle)

Side Wall King Common Rafter Slope Angle = arctan (tan  23.20706° ÷ sin 75.96375°) = 23.84263°

Side Wall King Common Rafter Run = (0.5 x Side Wall Length) x tan ( Plan Angle) 

Side Wall King Common Rafter Run = 16.97056" x tan ( 75.96375°) = 67.88222"

Side Wall King Common Rafter Length =  67.88222" ÷ cos ( 23.84263°) = 74.21586" TL

To calculate the actual Framing Length of the Side Wall King Common Rafter you need to calculate the dimension from the edge of the hip rafter at the plate line to the center of the side Wall King Common Rafter Run.

Side Wall Roof Sheathing Angle = arctan (tan 75.96376° ÷ cos 23.84263°) = 77.11992°

Hip Rafter Width ÷ sin(75.96376°) = 1.54616

33.94113" - 1.54616" = 32.39496"

32.39496" ÷ 2 = 16.19748"

Side Wall King Common Rafter Length = 16.19748" x tan  ( 77.11992°) = 70.83506" FL

For the Cheek Cuts on the Side Wall King Common Rafter

Side Wall King Common Plumb Line Shift

(Rafter Width x 0.5) x tan (Plan Angle)

0.75 x tan (75.96376°) = 3.00"

Mark the plumb cut on the rafter and measure perpendicular to the plumb line using the Plumb Line Shift dimension and mark the second plumb line on the side of the rafter. Set the saw blade bevel angle to the plan angle to cut the rafter from the side of the rafter. Or set your saw to 90° - plan angle to run the saw down the face of the rafter plumb cut.

The head cuts on the Bay Window hip rafters always present a problem with the cheek cut angles. Sometimes it's best to draw out the ridge connections full scale to transfer the shifted plumb lines  to the sides of  the hip rafter, so you can mark off the back bevel angles of the cheek cuts to determine the angles. Then there's the problem of the length of the hip rafter cheek cuts. I've use my swing table, 10" blade, to cut the cheek cuts or you can use a BigFoot or Beam Saw to cut the 3" to 4" long cheek cuts. Most of the time I'll cut the hip rafter cheek cut on the plumb face of the hip rafter. Set your saw to 90° - plan angle and cut the hip rafter  down the face of the plumb cut.

Saw Blade Bevel Angle on Side Of Hip Rafter 59.03625°... plan angle

Saw Blade Bevel Angle on Face Of Hip Rafter 30.96376°... 90° - plan angle

Saw Blade Bevel Angle on Side Of Hip Rafter 64.0664°.... planing angle of the 2 hip rafters in plan view.

Saw Blade Bevel Angle on Face Of Hip Rafter 25.93360°... 90° - planning angle

The HAP at the foot of the hip rafter also presents a problem with the unequal heel heights on each side of the hip rafter for roof plane alignment, since we're not using hip rafter offset/shift at the foot of the hip rafter. Most of the time the heel height difference is less than an 1/8". So it best to use the front wall heel height for the roof plane alignment height.

For the California Bay Window Hip Rafter you need to install the valley sleeper first. The edge bevel of the valley sleeper can be found from using geometry.

Or you can use trigonometry.

Roof Sheathing Angle = arctan (tan 45° ÷ cos 26.56505°)
48.18968° = arctan (tan 45° ÷ cos 26.56505°)

Valley Sleeper Miter Angle at the Top of the Sleeper = Jack Rafter Side Cut Angle = 41.81031°
Valley Sleeper Miter Angle at the Foot of the Sleeper = Roof Sheathing  Angle = 48.18968°
Valley Sleeper Saw Blade Bevel Angle = arctan(tan(90° - Main Roof Pitch°) x cos(Main Roof Sheathing Angle))
Valley Sleeper Saw Blade Bevel Angle = arctan(tan(90° - 26.56505°) x cos(48.18968°)) = 53.13010°

California Bay Window Hip Rafter Planning Point on Valley Sleeper. For the length of the California Bay Window Hip Rafter use my Rafter Tools app or pull out your tape measure and measure it.... Way too much math to explain here on the internet for this California Bay Window Hip Rafter length.

For the Purlin Miter & Back Bevel angles you can use geometry or trigonometry.

Example using geometry for the Purlin Miter angle. The Purlin Back Bevel Angle on the top edge of the purlin is the same as the Jack Rafter Side Cut angle. Or sometimes referred to as the top cut angle in Timber Framing.

Example using the geometry of a tetrahedron for the Purlin Miter angle, Purlin Back Bevel Angle and the Purlin Saw Blade Bevel Angle.

Example using Trigonometry to calculate the purlin miter & saw blade bevel angles.

Purlin Miter Angle = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Purlin Miter Angle = arctan( sin ( 26.56505 ) ÷ tan( 45 ))= 24.09484° 

Purlin Saw Bevel Angle = arcsin( cos ( Pitch Angle ) x cos( Plan Angle ))
Purlin Saw Bevel Angle = arcsin( cos ( 26.56505 ) x cos( 45 )) = 39.23152°

The frieze block angles for frieze block #5 are the same as the purlin angles, because of the 90° eave angle and the equal pitched roof.
Purlin = Frieze Block = Square Tail Fascia = Crown Molding Angles

Frieze Block # 1 & 4

Purlin Miter Angle = arctan( sin ( 26.56505 ) ÷ tan( 59.03624° ))= 15.02025° 

Purlin Saw Bevel Angle = arcsin( cos ( 26.56505 ) x cos( 59.03624° )) = 27.39866°

Frieze Block # 2 & 3

Purlin Miter Angle = arctan( sin ( 23.84263° ) ÷ tan( 75.96376°° ))= 5.77051° 

Purlin Saw Bevel Angle = arcsin( cos ( 23.84263° ) x cos( 75.96376°° )) = 11.36991°

Dog leg valley run = overhang_run /cos(90-main_plan_angle)
Dog leg valley run = 24 /cos(90-59.03624) = 27.98857

Dog leg valley slope angle = atan(overhang_rise / Dog leg valley run)
Dog leg valley slope angle = atan(12 / 27.98857) = 23.20706

For the Purlin Lip Cut Angle

Purlin Lip Cut Angle = arctan( tan (Hip Rafter Backing Angle) x cos( Jack Rafter Side Cut Angle )) Purlin Lip Cut Angle = arctan( tan ( 18.43495 ) x cos( 41.81032 )) = 13.95274°

The geometry for the purlin lip cut is the same as the purlin claw. The only difference is the height of the material above the purlin claw.

Purlin Claw

Hip Rafter Diamond Post Miter Angle = arctan( tan( hip rafter slope angle ) x cos (plan angle)) =14.03624°

Hip Rafter Diamond Post Saw Blade Bevel Angle = arctan( sin( Diamond Post Miter Angle ) x tan (Plan Angle))= 13.63302°

Jack Rafter Lengths

I like to calculate all of my Jack Rafter Lengths using the Roof Sheathing Angle. First you need to calculate the Hip Rafter Offset Along the Eave Line. With equal pitched roofs, on a 90° eave angle, the Hip Rafter Offset Along the Eave Line is the same dimension for all roof pitches. 

.

Hip Width x cos (45°) = Hip Offset Along Eave Line

3.5" x cos (45°) = 2.4749"

Hip Offset Along Eave Line = 2.4749"

First Jack Rafter Length

Jack Rafter Spacing + (1/2 of Jack Rafter Width)  - Hip Offset Along Eave Line = First Jack Rafter Run

First Jack Rafter Run x tan( Roof Sheathing Angle ) = First Jack Rafter Length

22.2751" x tan(48.18968°) = 24.9043"

Jack Rafter Length Deduction for Hip Rafter

Hip Offset Along Eave Line x tan(Roof Sheathing Angle) =  Jack Rafter Length Deduction for Hip Rafter

Hip Offset Along Eave Line = 2.4749"

2.4749" x tan(48.18968°) = 2.7670"

Jack Rafter Length Deduction for Hip Rafter = 2.7670"

King Common Rafter Length = 199.0101" TL

King Common Rafter Length To Hip Rafters = 199.0101" - 2.7670"  = 196.2431" FL

First Jack Rafter Length From King Common

23.25 x tan(48.18968°) = 25.9943"

196.2431" - 25.9943" = 170.2488" FL

Theoretical Length of Jack Rafter Slider "C"

Jack Rafter Run ÷ cos(Roof Slope Angle)

66" ÷ cos(26.56505°) = 73.79024" TL

Framing Length of Jack Rafter Slider (Doppelschifter )

Theoretical Length of Jack Rafter Slider - (Hip Rafter Deduction on Real Roof Surface x 2)

73.79024 - (2 x 2.7670") = 68.2563" FL for Slider "C"

Theoretical Length of Jack Rafter Slider D

Jack Rafter Run ÷ cos(Roof Slope Angle)

69.4359"" ÷ cos(26.56505°) = 77.63169" TL

Framing Length of Jack Rafter Slider

Theoretical Length of Jack Rafter Slider - (Hip Rafter Deduction on Real Roof Surface x 2)

77.63169 - (2 x 2.7670") = 72.09769" FL for Slider D

For the King Common Jack Rafter Length

Jack Rafter Slider Run ÷ cos ( Roof Slope Angle) =  King Common Jack Rafter Length TL

66 ÷ cos(26.56505°) = 73.79024" TL

King Common Jack Rafter Length TL - Hip Rafter Deduction on Real Roof Surface

73.79024 - 2.7670" = 71.02324" FL

Jack Rafter Length Difference = 26.83281"

Jack Rafter "A" Length =

71.02324" - 26.83281"  = 44.190432" FL

Using Geometry on the Real Roof Surface for the length of Jack Rafter "A"

For Jack Rafter "B"

For the King Common Jack Rafter Length

Jack Rafter Slider Run ÷ cos ( Roof Slope Angle) =  King Common Jack Rafter Length TL

69.4359 ÷ cos(26.56505°) = 77.63169" TL

King Common Jack Rafter Length TL - Hip Rafter Deduction on Real Roof Surface

77.63169 - 2.7670" = 74.86469" FL

Jack Rafter "B" Length Trigonometry???

It's better to layout Jack Rafter B using geometry or layout the roof full scale.

Hip Rafter Offset/Shift Examples

Hip Rafter Offset/Shift

The Hip Rafter Offset/Shift aligns the edges of the hip rafter material with the roof planes which are determined by the given Roof Slope Angles. This dimension is perpendicular to the hip rafter run line. You will layout the top of the unbacked hip rafter with these dimensions. 

In these example drawings the Eave angle is 90° and the two roof pitches are 8:12 & 10:12. The width of the hip rafter 3.5".

Hip Shift = Gratgrundverschiebung in German.

In these example drawings the Eave angle is 120° and the two roof pitches are 8:12 & 10:12. The width of the hip rafter 5.5".

In these example drawings the Eave angle is 135° and the roof pitches are equal @ 10:12. The width of the hip rafter 3.5".

In these example drawings the Eave angle is 90° and the roof pitches are equal @ 10:12. The width of the hip rafter 1.5".

Roof Framing Books From The Past

A couple of weeks ago I was asked what are some good books on Roof Framing. At that time I really couldn't recommend any particular American book on Roof Framing. After looking thru

Radford's cyclopedia of construction: 

carpentry, building and architecture, based on the practical experience of a large staff of experts in actual construction work, Volume 8 (Google eBook) printed in 1903
authors, William A. Radford, Alfred Sidney Johnson


It's probably on of the best book on Roof Framing in the last 100 years.It has resources  for finding the plan angles for unequal pitched roofs with a framing square and finding the back bevel cuts for hip and jack rafters using a form of shiften from plan view.

Hexenschnitt -- The Witches Cut, Square Tail Fascia
or
Co-Pitch

Radford has several drawings in the book that deal with square tail fascia. He uses the Co-Pitch to find the hip rafter miter angle. Radford's drawing look a lot like the German method that I use. Or the German method looks a lot like Radfords method.

Preface from the book:
Open Ways to Knowledge. — Of late years a number of books have been published, in which the authors and compilers have made commendable efforts to simplify matters pertaining to the arts of carpentry and joinery, and the mechanic of today has not the difficulties of his predecessors to contend with. The workman of old could excuse his ignorance of the higher branches of his trade, by saying that he had no means of acquiring a knowledge of them. Books were beyond his reach, and trade secrets were guarded so jealously, that only a limited few were allowed to know them, and unless he was made of better stuff than...

Radford's Cyclopedia of Construction on Google eBooks

Here's a couple of drawings I reproduced from Radfords drawings in his book on the Jack Rafter Shift Dimension in Profile for different polygon plan angles.The jack rafter shift dimension is different for each of the polygon plan angle, but the dimension stays the same for all of the different roof pitches for each type of polygon. 

This drawing shows the Plumb Line Shift Dimension for:

1. Square Pyramid (4 sides) = 90°
2.  Pentagon (5 sides) = 108°
3.  Hexagon (6 sides) = 120°
4.  Octagon (8 sides) = 135°
5.  Decagon (10 sides) = 144°
6.  Dodecagon (12 sides) = 150°
7.  Hexadecagon (16 sides) = 157.5°

Drawing of the Jack Rafter Plumb Line Shift Dimension for Equal Sided Octagon Roofs.

Roof Framing Book Table of Contents

Holy Grail of Roof Framing 
Geometry & Trigonometry

Complex Roof Framing Simplified

Trade secrets from Euclid, Cistercian monks, French Compagnons ,German Zimmermann and the American Carpenters & Builders of centuries past that only a limited few were allowed to know.

Possible Table of Contents
-----------------------------------------------------------------------

Developed Geometry From The Past

Finding the Lines of Descriptive Geometry by Monge

Folding Roof Surfaces

Hip Rafter Roof Plane Alignment

HAP-- Height Above Plate

Roof Framing Kernels

Prism Plane Geometry - Unbacked Shoulder of Hip Rafter

Understanding Plan Angles

Understanding Tangent Lines

Hip Rafter Offset/Shift

Hip Rafter Backing Angles -- Equal & Unequal Pitched Roofs

Hip Rafter Dihedral Angle for Hip Rafter Backing Angle

Back Bevel Angles -- Top Cut Angles

Hip Rafter Head Cuts -- Equal & Unequal Pitched Roofs

Jack Rafter Head Cuts -- Equal & Unequal Pitched Roofs

Using a Framing Square As A Protractor

Framing Square Usage for Laying out Rafters

Pyramid Hip Roof Rafters -- Equal & Unequal Pitched Roofs

Hexenschnitt -- The Witches Cut, Square Tail Fascia

Shed Roofs

Shed Roofs with Sloping Ridges

Shed Roofs with Walls not Parallel to the Ridge

Equal Pitched Gable Roofs

Unequal Pitched Gable Roofs

Unequal Pitched Gable Rafters with Unequal Plate Heights

Lapping Rafters Ridge Cut Angles for Equal & Unequal Pitched Roofs

Pitch To Plate Roof Trigonometry

Trapezoid Ground Plan Roof  with Unequal Overhang

Trapezoid Ground Plan Roof  with Equal Overhang

Octagon Hip Rafters

Octagon Roof Framing 4 Butt Hip Rafter Method

Octagon Roof Framing 4 Butt Hip Rafter Method with Purlin Structural Ring

Octagon Roof Framing with Gazebo Cupola

Hexagon Hip Rafters

Pentagon Hip Rafters

Polygon Roof Framing Angles

Curved Elongated Hexagon Roof Framing

Curved Elongated Octagon Roof Framing

Cupola Roofs

Snub Nose Hips

Clipped Hip Rafters

Dutch Hip Roofs

Tudor Roofs

Broken Hip Rafters

Blind Valley Rafters

Hidden Valley Rafters

Scotch Valley Rafters

Hip Dormer

Gable Dormer

Shed Roof Dormers

Drag Dormer
Schleppdachgaube

Triangular Dormer
Dreiecksgaube

Sloping Ridge Dormers

Drag Hipped Dormer

Abgewalmte Schleppdachgaube

Trapezoidal Dormer

Trapezgaube

CutIn Shed Roof Dormers

Barrel Roofs with Purlins

Barrel Roofs with Ribs

Eyebrow Roofs with Purlins

Eyebrow Roofs with Ribs

Circular Roofs

Octagon Foot Print on Circular Roofs

Elliptical Roofs

Ski Slope Rafters

Gambrel Roof Framing Angles

Dutch Gambrel Roof

Pent Roofs with Curved Rafters
Pultdach

Prow Roof Angles

Unequal Pitched Octagons

Unequal Sided Octagon Geometric & Trigonometric Roof Framing Development

Bay Window Rafters with Unequal Overhang

Bay Window Rafters with Equal Overhang

Bay Window Dog Leg Valley Rafters

California Valley Sleeper Bevel Cuts

Valley Sleeper Saw Blade Bevel Angle Geometric Development

Purlin Angles

Purlin Lip Cut Angles

Purlin or Dormer Square Tail Fascia Layover Cuts

Frieze Block Angles

Segmental Arch

Tudor Arch 
Ogee Arch 
Rampant Arch 
Lancet Arch 
Gothic Arch 
Morrish Arch 
Semicircle Arch 
Segment Arch 
Pointed Arch 
Elliptical Arch 
Equilateral Arch 
Double Curvature Arch 
Trompe Arch 
Catenary Arch

Lofting Equal Height Ordinates for Arches to Ellipses

Groin Vault Plotting -- Lofting Equal Height Ordinates for Elliptical Cross Vault

Semi-Circular Square Groin Vault
Semi-Circular Rectangular Groin Vault
Segmental Square Groin Vault
Segmental Rectangular Groin Vault
Gothic Square Groin Vault
Gothic Rectangular Groin Vault
Semi-Circular Hexagon Groin Vault
Segmental Hexagon Groin Vault
Gothic Hexagon Groin Vault
Semi-Circular Octagon Groin Vault
Segmental Octagon Groin Vault
Gothic Octagon Groin Vault
Semi-Circular Dodecagon Groin Vault
Segmental Dodecagon Groin Vault
Gothic Dodecagon Groin Vault
Semi-Circular Hexadecagon Groin Vault
Segmental Hexadecagon Groin Vault
Gothic Hexadecagon Groin Vault

Crown Molding Angles --> Square Tail Fascia -- Frieze Block -- Purlin Angles

Hopper Angles

Creeper Rafters

Plumb bevel --> Level Bevel --> Edge Bevel
Edge bevel creeper
Edge bevel purlin
Face bevel purlin

Plywood Cuts

Hip Rake Walls Rotated into Roof Surface Plane

Hip Rafter Diamond Post Calculations

Alhambra Granada Ad Quadratum Ground Plan

Octagonal Baptismal

Joe Bartok Roof Framing Studies

Complex Roof Framing by Billy Dillon

Unequal Pitched Valley Rafter Framing by Tim Uhler

One Length Method© by Richard Birch Irregular Hip & Valley Rafters

The Cube Method

Framing Square Usage for Plan Angles

Simplified Roof  Framing

Timber Framing

Fourteen Working Planes of  Roof Framing Geometry

Log Joinery Framing

Traditional Roof Framing Geometry Layout

Shiftungen Technique

Traité de Charpente, Art of Line

Rabattement Surface, Folding Plane or Folding Roof Surface

Drawn Down Method

Devers de Pas

Reciprocal Roof Framing  Geometry

Prismatic Foot Print

Rafters Rotated Plumb To Earth

Rafters Rotated Plumb To Roof Surface

Skewed Rafters Rotated Plumb To Earth

Skewed Rafters Rotated Plumb To Roof Surface

Saint Andrews Cross

Claw Angles

Stereotomic & Descriptive Geometry
Stereotomy (masonic projection) techniques

Spiral - Helix  - Circular Stair Geometric Development

Tangent Handrailing & the Prism Plane

Platonic Solids

Trigonometry Formulas

Unit Circle

Circle Tangent To Hip Rafter

Law of Sines

Law of Tangents

Law of Cosines

Cosine Trigonometry Formulas

Law of Sines Trigonometry Formulas

Pythagorean Trigonometric Identities

Tetrahedrons

Tetrahedron Slices Folded & Unfolded

Canadian & American Geometric Roof Framing Development & Framing Square Usage
Compound Miter Angle and Saw Blade Bevel Angles

Saw Blade Angle Development

Direction Of Saw Travel

Bartok, Hawkindale, Martindale and McKibben-Gray Hip-Valley Roof Ratios Angle Formulas.

Bevel angles for three dimensional connections

Euclid's Elements

Euclid's 47th Proposition

Pythagorean Theorem

Thales’ Theorem to Construct a Right Triangle

Formula for a golden ratio ellipse
Formula for a golden ratio stairs
Formula for a golden ratio Roof slope Angle

Vitruvian Man

Conic Sections
conic section, ellipse, parabolas, hyperbolas

Solid Geometry
Horizontal Plane
Vertical Plane
Horizontal Trace of Oblique Plane
Vertical Trace of Oblique Plane

Planar Geometry

Circle Geometry

Umbra Recta
Orthography and Scenography

WorldSkills Carpentry Competition Training

Japanese Carpentry

trait de charpente

Euclid

Zimmerman

Compagnonnage
FreeMasons masonic projections technique
Cistercian monks -> French Compagnons ->Carpenters’ masterworks, twisting and turning beyond the geometrical kernel

Ad Quadratum - Daisy Wheel - Ad Triangulum

1. Sand Geometry
2. Light Geometry
3. Egyptian Geometry
4. Greek Geometry
5. Roman Geometry
6. Arabic/Islamic Geometry
7. Persian Geometry
8. Pythagoras Geometry
9. Euclidean Geometry 
10. Vitruvian Geometry
11. Archimedes Geometry
12. Apollonios Geometry
13. Sacred Geometry
14. Vesica Piscis Geometry
15. Ad Triangulum Geometry
16. √3 Geometry
17. Ad Quadratum Geometry
18. √2 Geometry
19. Golden Ratio phi
20. Star of David or Solomon's seal 
21. Thunder Mark's
22. Euclidean six point geometry
23. Empirical geometry
24. Descriptive Geometry 
25. Spherical Elliptical Geometry 
26. Equilateral triangle Geometry
27. Stone Cutting Geometry
28. Trompes(ribless, conical vaults)

Bibliography

Euclid of Alexandria (200 BC) Greek 
Archimedes of Syracuse (287 – 212 BC) Greek
Apollonius of Perga (262 – 190 BC) Greek 
Marcus Vitruvius Pollio (80 – 70 BC) Roman
Anthemius of Tralles (474 – 558) Greek - Roman
Abul Wafa al-Buzjani (940 – 998) Persian, Baghdad 
Villard de Honnecourt (1225) Cistercian Order of France
Matthäus Roritzer (1435 – 1495) German
Albrecht Dürer (1471 – 1528) German
Rodrigo Gil (1500 – 1577) Spanish
Andrea Palladio (1508 – 1580) French
Philibert De l'Orme, (1515 – 1576) French, Le premiere tome dell’architecture
Francois Derand (1588 – 1644) French 
Mathurin Jousse(1607 – 1692) French
Gérard Desargues(1591 – 1661) French 
Amédée-François Frézier(1682 – 1773) French
Gaspard Monge (1746 – 1818) French
Peter Nicholson (1765–1844) British 
Asher Benjamin (1773 – 1845) American 
Nicolas Fourneau (1722 – 1792) French
Louis Mazerolle (1800) French
JD Boucher (1800) French
Emile Detatille (1800) French
Billon Freres (1800) French
Robert Willis (1800 – 1875) British

Art de la charpenterie; by A. R. Emy. 1841-42
J. Newlands. Carpenters' and joiners' assistant; 1860

Hetsch Holzarchitectur des mittelalters; by C.Botticher. 1841

J.C. Krafft  1815

Traité sur l'art de la charpente théorique et pratique: Anweisung zur theoretisch-praktischen Zimmermanns-Kunst. Treatise on the art of carpentry, with the theory and practice

W. Pain,  Practical House Carpenter 1799

T. Tredgold, Elementary Principals of Carpentry 1821

J. Smith, Specimens of Ancient Carpentry 

Nicolai  Zabaglia 1743
Leonardo da Vinci
François Mansart
Gabriel-Philippe de LaHire
Charles-François-Antoine
Nicolas Fourneau
Peter Nicholson
Louis Mazerolle
JD Boucher
Emile Detataille
Cabanie
Robert Ridial
Ira Samuel Griffith
William A. Radford
Alfred W. Woods
G.C. Volland

G.C. Schulze
Armand Rose Emy Traitè de l’Art de la Charpenterie
J. B. Rondelet
Charles A. King
J. W. Riley
Simplified Roof Framing by J. D. Wilson and S. O. Werner 1927
A Roof Cutter's Secrets by Will Holladay 2006
Marie-Thérèse Zenner
Timber Framers Guild 

Nexus Network Journal
The Edinburgh Encyclopedia

Elementa Geometriae- 1482

Gilbert Emery

Jean-Michel Emery

Philippe Nairière

Marie-Paule Raimbault

Peter Kübler
Albert Müller
Andreas Großhardt
Hans Wittmann
Roland Schumacher
Michael Riggenbach

Manfred Euchner 

Von Franz Krÿ¤mer

Silas Hawes Framing Square 1814

Eagle Square Manufacturing Co. 1846

Howard square 1881

Sargent Square 1914
Nicholls Square 1901
Chappell Master Framing Square 2010



U.S. Patent Documents for Carpentry Squares

247353	September 1881	Howard
492532	February 1893	Gilmer
651057	June 1900	Roberts
672455	April 1901	Nicholls
691192	January 1902	Smith et al.
937202	October 1909	Bailey
1102689	July 1914	Sargent
1196519	August 1916	Caylor
1236817	August 1917	Bick
1241976	October 1917	Hill
1463605	July 1923	Walters
1477002	December 1923	Parkhill
1704462	March 1929	Crandlemere
2654954	October 1953	Lawrence
4200990	May 1980	West
4420891	December 1983	Orem
5727325	March 1998	Mussell
6105266	August 2000	Cote
6725555	April 2004	Moore
6868616	March 2005	Allemand
7854070	December 2010	Vajentic

------------------------------------------------------------------------------

550 Ausseur (J. J.) Traité de la Coupe de Bois, ou Art du

Trait du Menuisier en Bâtiment, 4to. 35 plates, 1/. 4s. Paris, 1819

551 Barlow's (P.) Essay on the Strength and Stress of

Timber, 8vo. plates, 16*. 1817

552 Crocker's (A.) Timber Dealer's Guide, 4s. 6d. 1813

553 Essai sur les Bois de Charpente, 12mo. 3*. 6d.

Paris, 1763

554 Emerson's Principles of Mechanics, plates, neat, 15s.

1773

555 Fourneau (Nie.) l'Art du Trait de Charpenterie, 4 tom.

folio, 87 plates, sewed, 4L 4s. Pans, 1820

556 Hassenfratz's (J. H.) Traité de l'Art du Charpentier,

4to. 28 plates, II. 10*. Pans, 1804

557 Jousse (M.) Art de Charpenterie, folio, plates, 1/. 10*.

Paris, 1751

558 Krafft, Plans, Coupes, et Elévations de diverses Pro

ductions de l'Art de la Charpente exécutées tout en France que dans les Pays Etrangers, large and thick folio, about 400 plates, neatly half-bound, 8/. 18*. 6d. Paris, 1805

559 Krafft's (J. C.) Treatise on the Art of Carpentry, with

the Theory and Practice, folio, 180 plates, 5/. 15*. 6d.

SCO Krafft's (J. C.) Carpentry (Part 1 of the Work), comprising, Des Assemblages, Traits de Jupiter, Armatures des Poutres, Courbs en Planches, des Soffites, Pans de Bois, Planchers, 30 plates, 1/. 5s. Paris

561 Krafft's (J. C.) Carpentry (Part 2 of the Work), comprising Des Escaliers, 30 plates, 1/. 5s. Paris

562 Krafft's (J. C.) Carpentry (Part 4 of the Work), comprising Des Combles, 30 plates, 1/. 5s. Paris

563 Krafft's (J. C.) Carpentry (Part 5 of the Work), comprising Des Combles, Clochers, Coupoles, &c. 30 plates, II. 5s. Paris

564 Krafft's (J. C.) Carpentry (Part 6 of the Work), Construction of Theatres, 30 plates, 21. 12s. 6d.

Paris, 1822

565 Nicholson's (P.) Carpenter and Joiner's Assistant, 4to.

bound, 1/. Is. 1815

566 Nicholson's (P.) Carpenter's New Guide, 84 plates,

567 Nicholson's (P.) Treatise on the Construction of

Staircases and Handrails, 4to. 39 plates, bound, 18s.

1820

568 Pain's (W.) Practical House Carpenter, 4to. 146

plates, 15s. 1799

569 Smith's (J.) Specimens of Ancient Carpentry, 4to. 36

plates, 12s.

570 Tredgold's (T.) Elementary Principles of Carpentry,

4to. 22 plates, 1/. 4s. ' 1821

571 Zabaglia (Nicolai) Contignationes ac Pontes, ac Descriptioncs translationis Obelisci Vaticani, large folio, 54 plates, and a full length portrait of the

One of the reasons for me choosing

The Holy Grail of Roof Framing Geometry and Trigonometry

 for the title of my book.

Research on my mother’s maiden name of St. Clair.

Rosslyn Chapel was a rebuilding of the Temple of Solomon, designed by St. Clair

Research shows that ras or ros means wisdom. As such, could it be that the term ‘roslin’ means the St. Clairs , who were also the guardians of the fountain of wisdom pertaining specifically to the Holy Blood, the Holy Grail and the Holy Light.

Craft  Guilds --> Knights of the Templar --> St. Andrews Cross --> St. Clair -->Rosslyn Chapel --> Temple of Solomon --> Fountain of Wisdom --> Holy Grail

Formulas arise from the geometry #2

1980’s version of the hip rafter backing angle formula I used. It was developed from the book 

Full Length Roof Framer by A.F.J Riechers

One triangle at a time type of formula

Hip rafter backing angle = arctan( ((hip rafter thickness * 0.5) * sin(hip rafter slope angle)) / (hip rafter thickness * 0.5))

It wasn't very efficient, but at least it was correct for equal pitched roofs.

Roof framing is the Ph.D. of carpentry, it's apparent that  Marshall Gross never got his Ph.D. of carpentry.

I was thumbing through Marshall Gross's book "Roof Framing", 1984, and on page 135 he said to use the hip rafter slope angle as the hip rafter backing angle to cut the hip rafter edge bevels. Not only was the hip rafter backing angle wrong, his hip rafter backing depth on the side of the hip rafter was also incorrect. No way was he ever taught by a German Master carpenter. A German Master Carpenter would have taught him how to layout the backing angles correctly for any roof by using geometry that was developed centuries ago. Hip Rafter Backing Angle Formulas are developed from geometry, not from sketches of what you think are correct.

Hip Rafter Backing Angles for Edge Bevel Formulas

Formulas arise from the geometry

Joe Bartok said Formulas arise from the geometry. This is the cornerstone of all my new/old trigonometry formulas after I meet Joe. Before I meet Joe I would draw 20 different triangles on the same piece of paper to calculate the roof angles. Now my preferred method is to first draw out a tetrahedron. If I can't find the angles I'm looking for in the tetrahedron, I'll take a tetrahedron slice cut from the rafter. Then I'll unfold the tetrahedron slice and finally I'll develop the correct tetrahedron showing the relationships of the different roof framing angles that I can use to develop the trigonometric formula.

Tetrahedron Slice

Miter Angle R2 -- Witches Cut -- Square Tail Fascia Miter Angle on the Side Of Hip Rafter

Saw Blade Bevel C1
Develops Bevel Angle 90 - R3 on the top edge of the hip rafter material.

Use C1 for the saw blade bevel angle along the miter line (R2) on the side of the hip rafter.

C1 = Saw Blade Bevel Angle = arcsin( cos ( Pitch Angle ) * cos( Plan Angle ))

or

Known Angles

1. R2
2. R3
3. C1
4. C2

C1 = Saw Blade Bevel Angle = arctan( cos ( R2 ) * tan( R3 ))

C1 = Saw Blade Bevel Angle = arcsin( sin ( R3 ) * cos( C2 ))

C1 = Saw Blade Bevel Angle = arccos( sin ( C2 ) ÷ sin( R2 ))

C1 = Angle B in the tetrahedron

Tetrahedron Trigonometric Identity Formulas

by Joe Bartok

Angle B = arccos( sin ( D ) ÷ cos ( E ))

Angle B = arccos( cos ( A ) ÷ cos ( C ))

Angle B = arcsin ( sin ( A ) × cos ( D ))

Angle B = arcsin ( tan ( E ) × tan ( C ))

Angle B = arctan( tan ( A ) × sin ( E ))

Angle B = arctan( sin ( C ) ÷ tan ( D ))

1. C1 = arccos( sin ( C5 ) ÷ cos ( 90 - P1 ))
2. C1 = arccos( cos ( P2 ) ÷ cos ( R2 ))
3. C1 = arcsin ( sin ( P2 ) × cos ( C5 ))
4. C1 = arcsin ( tan ( 90 - P1 ) × tan ( R2 ))
5. C1 = arctan( tan ( P2 ) × sin ( 90 - P1 ))
6. C1 = arctan( sin ( R2 ) ÷ tan ( C5 ))