Chappell Master Framing Square and the Purlin Lip Cut Angle

 What is the purlin lip cut angle? The purlin lip cut angle is on the upper shoulder of a roof purlin that allows the roof purlin to project under the hip or valley rafter at a  plane that is perpendicular to the hip or valley rafter plane.

In this post I'm using the  Chappell Master Framing Square to demonstrate how to lay out the rafters of an unequal pitched roof (bastard roof), major pitch 12:12 and a minor pitch of 9:12 with an eave angle of 90° and how to layout and cut the purlin lip cut angle.

The first thing we need to do is layout the hip rafter offset. The drawing below, that dates back to the 1800s or earlier, is the easiest way to lay out the hip rafter offset. The hip rafter in this drawing is 3.5" in thickness. Mark off a line that is perpendicular to the hip rafter run and the width of the hip rafter. Then draw a line that is parallel with the adjacent eave line to obtain the hip rafter offset. In this drawing the hip rafter offset for the 12:12 side is 1.26" and the hip rafter offset for the 9:12 side is 2.24"

Next lay out the hip rafter. The hip rafter angle is on the first line of the Chappell Master Framing Square under the 9:12 on the Unequal Pitched 12/12 Main Pitch A side of the Chappell Master Framing Square. 30.964°. To layout the hip rafter plumb line use the .6 on the square. Multiply it by 10, which gives us 6.0, and use 6" on the tongue and 10" on the blade. (6 ÷ 10 = arctan (0.6)= 30.96376°).

 Next mark off the hip rafter backing depth on each side of the hip rafter. The equal pitch rafter tables on the Chappell Master Framing Square on line 4, DEPTH OF BACKING & BEVEL CUT IN INCHES PER 1" OF HIP OR VALLEY WIDTH (Multiply 1/2 Beam Width by Factor Given for Total Depth) are the tangents of the Hip or Valley rafter backing angle. For the Hip rafter in our example the hip rafter backing angle for the 12:12 side is 34.450°. The tan of 34.450° is = 0.685997.  Multiply 0.685997 by 1.24", the hip offset, to find the depth of the hip rafter backing line along the side of the hip rafter.  0.685997 x 1.24 = 0.850636 or 7/8".  On the 9:12 side we'll take the tan of 21.10° (0.385868) then multiply it by  2.24. 0.385868 x 2.24 = 0.864344 or 7/8".

To double check the hip rafter backing depth marked on each side of the hip rafter you can draw out the hip rafter plumb backing angles on the end of the rafter. 

9:12 side
Hip Rafter Backing Angle = arctan( sin( Hip Rafter Pitch Angle) ÷ tan( Plan Angle ) )
Hip Rafter Backing Angle = arctan( sin( 30.9637579168711) ÷ tan( 53.1301 ) ) = 21.1001984464996°
Hip Rafter Plumb Backing Angle = arctan( tan( Pitch Angle ) * cos( Plan Angle ))
Hip Rafter Plumb Backing Angle = arctan( tan( 36.8699 ) * cos( 53.1301 )) = 24.2277483279458°


12:12 side
Hip Rafter Backing Angle = arctan( sin( Hip Rafter Pitch Angle) ÷ tan( Plan Angle ) ) 
Hip Rafter Backing Angle = arctan( sin( 30.9637579168711) ÷ tan( 36.8699 ) ) = 34.4499007767502°
Hip Rafter Plumb Backing Angle = arctan( tan( Pitch Angle ) * cos( Plan Angle ))
Hip Rafter Plumb Backing Angle = arctan( tan( 45 ) * cos( 36.8699 )) = 38.6598073928135°

In this picture we can see that the hip rafter backing mark on the 9:12 side of the hip rafter aligns with the hip rafter plumb backing  angles on the end of the hip rafter plumb cut.

Next draw out the hip rafter side cut angles at the foot of the hip rafter.  The hip rafter side cut angles are the opposite of the hip rafter side cut angles at the peak of the hip rafter. In this drawing you can see how the hip rafter side cut angles at the foot of the hip rafter align with the eave lines and the hip rafter backing depth mark.

Hip Rafter Side Cut Angle at Peak  = arctan( cos( Hip Rafter Pitch Angle ) ÷ tan( Plan Angle ))

Hip Rafter Side Cut Angle at Foot  = arctan( cos( Hip Rafter Pitch Angle ) ÷ tan( Adjacent Plan Angle ))

12:12 side

Hip Rafter Side Cut Angle at Peak = arctan( cos( 30.9637579168711 ) ÷ tan( 36.8699 )) = 48.825665948811°Hip Rafter Side Cut Angle at Foot = arctan( cos( 30.9637579168711 ) ÷ tan( 53.1301)) = 32.7458710613747°

9:12 side

Hip Rafter Side Cut Angle at Peak = arctan( cos( 30.9637579168711 ) ÷ tan( 53.1301 )) = 32.7458710613747°Hip Rafter Side Cut Angle at Foot = arctan( cos( 30.9637579168711 ) ÷ tan( 36.8699)) = 48.825665948811°

The length of the hip rafter is calculated by multiplying the decimal on line 3 of the square by the run of the common rafter. In this model the run of the 12:12 common rafter is 16.3125". So we multiply 16.3125 x 1.944 = 31.7058" or 31 11/16" for the plumb to plumb length of the hip rafter.

The length of the jack rafters are calculated by multiplying the decimal on line 4 of the square by the length of the eave line of the common rafters. In this model the eave line length from the hip rafter offset line for the  12:12 common rafter side  is 19.65". So we multiply 19.65 x 1.061 = 20.8420" or 20 13/16" for the jack rafter length on the 12:12 side. In this model the eave line length from the hip rafter offset line for the  9:12 common rafter side  is 13.5125". So we multiply 13.5125 x 1.667 = 22.5208" or 22 1/2" for the jack rafter length on the 9:12 side. 

Note: The values on line 4 of the Chappell Master Framing Square are the tangents of the roof sheathing angles. The tan of the roof sheathing angle times the jack rafter spacing, equals the difference in length of the jack rafters.  If the jack rafters were spaced 24" O.C. for the 12:12 side of the roof, then we would use 24" x 1.061 = 25.45589" for the difference in length of the jack rafters for the 12:12 side of the unequal pitched roof.

Next cut the purlin's using Line 7 on the Chappell Master Framing Square for the purlin miter angle on the side of the purlin. This is where the Chappell Master Framing Squares are worth the hefty price of the square. It's one thing to have the purlin miter angle and use a speed square to layout the purlin miter angle, but by using the decimal on line 7 of  the square, .9428 in this example, all you have to do is multiply it by 10 and use 10" on the blade of the square to layout angles accurately. No more calculator calculations to transfer the angles to a framing square for accuracy.  For the cut on top of the purlin you can use line 5 on the Chappell Master Framing Square and the hip rafter backing bevel on line 7. 

The purlin lip cut angle is P3 of the Hawkindale angles.
P3 = Arctan (Cos D x Sin R1 x Cos R1 ÷ Cos S)

http://www.tfguild.org/tools/hipart.html


Calculations for the purlin lip cut angle for the 12:12 side of roof
Hip Rafter Square Tail Side Cut Angle (Foot) = 52.13635
R3 = 90° - Hip Rafter Square Tail Side Cut Angle (Foot) = 37.86365
C2 = Hip Rafter Square Tail Side Cut Bevel Angle (Foot) = 22.83365

R3 equals P2 rotated by C5.
R3 equals Jack Rafter Side Cut Angle rotated by the Hip Rafter Backing Angle
R3 = arctan( cos ( Hip Rafter Backing Angle ) * tan( Jack Rafter Side Cut Angle ))
R3 = arctan( cos ( 34.4499007767502 ) * tan( 43.3138542102836 )) = 37.8636443877934°

Tetrahedron showing the relationships of the Hip Rafter Back Angle, Jack Rafter Side Cut Angle, 90° - Hip Rafter Square Tail Side Cut Angle (Foot), Hip Rafter Square Tail Side Cut Bevel Angle (Foot)  and the Purlin Lip Cut Angle.

Purlin Lip Cut Angle = P3
P3 = arccos( sin (  Jack Rafter Side Cut Angle rotated by the Hip Rafter Backing Angle  ) ÷ sin ( Jack Rafter Side Cut Angle ))
P3 = arccos( sin ( R3 ) ÷ sin ( P2 ))
P3 = arccos( sin ( 37.86373 ) ÷ sin ( 43.3139 ))
Purlin Lip Cut Angle = arccos( sin (  90° - Hip Rafter Square Tail Side Cut Angle (Foot)  ) ÷ sin ( Jack Rafter Side Cut Angle ))

An easier  Purlin Lip Cut Angle Formula by Joe Bartok.
Purlin Lip Cut Angle = arctan (tan Backing Angle × cos Jack Rafter Side Cut Angle)

Purlin with lip cut angle.

You can draw the purlin lip cut angle on the side of the purlin. However, all you need to do is run your skill saw down the face cut of the purlin to produce the purlin lip cut angle. The purlin lip cut angle follows the bottom shoulder of the hip rafter. If you draw a line perpendicular to the hip rafter triangle, it will trace the purlin lip cut angle on the purlin, because the purlin face cut, once installed, is a vertical plane that matches the hip rafter triangle plane .

Purlin with Lip Cut Angle Drawn on the side of the Purlin.

12:12 & 9:12  purlin's with purlin lip cut angle.

Here's a better picture, by Joe Bartok,  of the Purlin Lip Cut Angle (P3) showing the square cut along the face cut of the purlin, with zero blade bevel, to produce the purlin lip cut angle that is perpendicular to the hip rafter triangle plane.

Here's two more drawings by Joe Bartok showing the relationship of P3 , P2 and C5 in a tetrahedron.

sin(90° - P2) = cos(P2)

C5 = Hip Rafter Backing Angle

P2 = Jack Rafter Side Cut Angle

Angle C = arctan( tan ( A ) × cos ( D ))

or

Angle P3 = arctan( tan ( C5 ) × cos ( P2 ))

Purlin Lip Cut Angle = arctan( tan ( Hip Rafter Backing Angle ) × cos ( Jack Rafter Side Cut Angle  ))

Roof Framing kernel with geometric layout for the purlin miter angle and purlin lip cut angle.

Chappell Master Framing Square Unequal Pitched Roof Review

In this blog/article I'm taking the Chappell Master Framing Square for a test drive for accuracy using the Unequal Pitched Roof Rafter Tables imprinted on the stainless steel carpenters framing square.












We'll use an Eave angle of 90° with an 12:12 major pitch (B)and an minor pitch of 9:12 (A).

This drawing is a French Compagnon Roof Framing Kernel drawn in 2D and 3D. The 2D geometric drawing of the French Compagnon Roof Framing Kernel is two tetrahedrons unfolded. Checking the 2D & 3D geometric drawing angles against the Chappell Master Framing Square clearly shows how accurate the unequal pitched roof rafter tables are on the Chappell Master Framing Square. This unequal pitched roof is an 12/12 and 9/12 with an eave angle of 90°.

Major Pitch 12/12
Minor Pitch 9/12
Major Plan Angle = 36.86990° tan: 0.75
Minor Plan Angle = 53.13010° tan: 1.333333
Major Pitch Angle = 45.00000° 12/12
Minor Pitch Angle = 36.86990° 9/12
Hip Rafter Angle = 30.96376° tan: 0.6 (line 1 on the square)
Major Hip Rafter Backing Angle = 34.44990° (line 9 on the square)
Minor Hip Rafter Backing Angle = 21.10020° (line 9 on the square)
Major Hip Rafter Sidecut Angle = 48.82567° tan: 1.143324
Minor Hip Rafter Sidecut Angle = 32.74587° tan: 0.64312
Major Jack Rafter Sidecut Angle = 43.31386° tan: .942809 (line 4 on the square)
Minor Jack Rafter Sidecut Angle = 30.96376° tan: 0.6 (line 4 on the square)
Major Roof Sheathing Angle = 46.68614° tan: 1.06066 (line 5 on the square)
Minor Roof Sheathing Angle = 59.03624° tan: 1.666666 (line 5 on the square)
Major Hip Rafter Square Tail Saw Bevel Angle = 34.44990° (line 9 on the square)
Minor Hip Rafter Square Tail Saw Bevel Angle = 28.685° (line 9 on the square)
Major Hip Rafter Square Tail Miter Angle = 28.07248° tan: 0.5333 (line 8 on the square)
Minor Hip Rafter Square Tail Miter Angle = 12.18863° tan: 0.216 (line 8 on the square)
Major Purlin Miter Angle = 43.31385° tan: 0.942809 (line 7 on the square)
Minor Purlin Miter Angle = 24.22774° tan: 0.45 (line 7 on the square)

The only roof framing angle missing on the Chappell Master Framing Square is the hip rafter side cut angle. However, you can set your saw blade bevel angle to the plan angle of the roof pitch to produce the side cut angle on the hip rafter.

This drawing is a tetrahedron slice of the hip rafter side cut angle at the ridge for the 12/12 side of the hip rafter.

 Major Pitch 12/12
 Minor Pitch 9/12
 Major Plan Angle = 36.86990° tan: 0.75
 Major Pitch Angle = 45.00000° 12/12
 Hip Rafter Angle = 30.96376° tan: 0.6
 Major Hip Rafter Backing Angle = 34.44990°
 Major Hip Rafter Sidecut Angle = 48.82567° tan: 1.143324

This drawing is a tetrahedron slice of the hip rafter side cut angle at the ridge for the 9/12 side of the hip rafter.

 Major Pitch 12/12
 Minor Pitch 9/12
 Minor Plan Angle = 53.13010° tan: 1.333333
 Minor Pitch Angle = 36.86990° 9/12
 Hip Rafter Angle = 30.96376° tan: 0.6
 Minor Hip Rafter Backing Angle = 21.10020°
 Minor Hip Rafter Sidecut Angle = 32.74587° tan: 0.64312


Note: All drawings in this post were generated using my Google Sketchup Roof Framing Geometry plugin.

Checking the trigonometric calculations below against the Chappell Master Framing Square clearly shows how accurate the unequal pitched roof rafter tables are on the Chappell Master Framing Square. Whether your a stick framer or a timber framer.
Tetrahedron Angles for 12/12 irregular hip roof

D Angle = 36.8699
A Angle = 45.0000
C Angle = 30.9638
E Angle = 43.3139
B Angle = 34.4499

90-D Angle = 53.1301
90-A Angle = 45.0000
90-C Angle = 59.0362
90-E Angle = 46.6861
90-B Angle = 55.5501

The dihedral angle is the angle measured between two planes.
Dihedral Angle = 111.1002

Polygonal Pyramid Sides = 3.39
Pyramid Slant Angle = 45
Pitch Angle = 45
Plan Angle  = 36.8699
Working Angle = 53.0973451327434

Hip Rafter Pitch Angle = arctan( tan( Pitch Angle ) * sin( Plan Angle ))
Hip Rafter Pitch Angle = arctan( tan( 45 ) * sin( 36.8699 )) = 30.9637579168711°

Hip Rafter Backing Angle = arctan( sin( Hip Rafter Pitch Angle) ÷ tan( Plan Angle ) ) 
Hip Rafter Backing Angle = arctan( sin( 30.9637579168711) ÷ tan( 36.8699 ) ) = 34.4499007767502°

Hip Rafter Plumb Backing Angle = arctan( tan( Pitch Angle ) * cos( Plan Angle ))
Hip Rafter Plumb Backing Angle = arctan( tan( 45 ) * cos( 36.8699 )) = 38.6598073928135°

Hip Rafter Side Cut Angle  = arctan( cos( Hip Rafter Pitch Angle ) ÷ tan( Plan Angle ))
Hip Rafter Side Cut Angle = arctan( cos( 30.9637579168711 ) ÷ tan( 36.8699 )) = 48.825665948811°

Jack Rafter Side Cut = arctan( cos( Pitch Angle ) ÷ tan( Plan Angle ))
Jack Rafter Side Cut = arctan( cos( 45 ) ÷ tan( 36.8699 )) = 43.3138542102836°

Roof Sheathing Angle = arccos( cos( Plan Angle ) * cos( Hip Rafter Pitch Angle ))
Roof Sheathing Angle = arccos( cos( 36.8699 ) * cos( 30.9637579168711 )) = 46.6861457897164°

Frieze Block Saw Miter Angle = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Frieze Block Saw Miter Angle = arctan( sin ( 45 ) ÷ tan( 36.8699 )) = 43.3138542102836°

Frieze Block Saw Blade Bevel Angle = arctan( sin( Frieze Block Saw Miter Angle ) ÷ tan( Pitch Angle ))
Frieze Block Saw Blade Bevel Angle = arctan( sin( 43.3138542102836 ) ÷ tan( 45 )) = 34.4499007767502°

Square Tail Fascia Miter Angle  = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Square Tail Fascia Miter Angle  = arctan( sin ( 45 ) ÷ tan( 36.8699 )) = 43.3138542102836°

Square Tail Fascia Saw Blade Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Square Tail Fascia Saw Blade Bevel Angle  = arcsin( cos ( 45 )  *  cos( 36.8699 )) = 34.4499007767502°

Hip Rafter Square Tail Miter Angle  = arctan (( cos( Pitch Angle) * sin( Pitch Angle ) * cos(Plan Angle)) ÷ tan( Plan Angle))
Hip Rafter Square Tail Miter Angle  = arctan (( cos( 45) * sin( 45 ) * cos( 36.8699 )) ÷ tan(36.8699)) = 28.0724841662578°

Hip Rafter Square Tail Saw Blade Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Hip Rafter Square Tail Saw Blade Bevel Angle  = arcsin( cos ( 45 )  *  cos( 36.8699 )) = 34.4499007767502°

90° - Hip Rafter Square Tail Side Cut Angle (Foot)
R3 equals P2 rotated by C5.
R3 equals Jack Rafter Side Cut Angle rotated by the Hip Rafter Backing Angle
R3 = arctan( cos ( Hip Rafter Backing Angle ) * tan( Jack Rafter Side Cut Angle ))
R3 = arctan( cos ( 34.4499007767502 ) * tan( 43.3138542102836 )) = 37.8636443877934°

Dormer Square Tail Fascia Miter Angle  = arctan( cos ( Dormer Pitch Angle ) ÷ tan( Main Roof Pitch Angle ))
Dormer Square Tail Fascia Miter Angle  = arctan( cos ( 45 ) ÷ tan( 45 )) = 35.2643896827547°
Dormer Square Tail Fascia Miter Angle on Fascia = 90° - 35.2643896827547° = 54.7356103172453°

Dormer Square Tail Saw Blade Bevel Angle  = arcsin( sin ( Dormer Pitch Angle ) *  cos( Main Roof Pitch Angle ))
Dormer Square Tail Saw Bevel Angle  = arcsin( sin ( 45 )  *  cos( 45 )) = 30°


Rake Stud Saw Miter Angle = Hip Rafter Plumb Backing Angle = 38.6598073928135°
Rake Stud Saw Blade Bevel Angle = arctan(cos(Hip Rafter Plumb Backing Angle)* tan(Hip Rafter Pitch Angle))
Rake Stud Saw Blade Bevel Angle = atan(cos(38.6598073928135)* tan(30.9637579168711)) = 25.1040917208509°

Rake Plate Miter Angle = Jack Rafter Side Cut Angle = 43.3138542102836°
Rake Plate Saw Blade Bevel Angle = Roof Pitch Angle = 45°


Purlin Miter Angle  = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Purlin Miter Angle  = arctan( sin ( 45 ) ÷ tan( 36.8699 ))= 43.3138542102836°

Purlin Saw Blade Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Purlin Saw Blade Bevel Angle  = arcsin( cos ( 45 )  *  cos( 36.8699 )) = 34.4499007767502°

Crown Molding Miter Angle  = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Crown Molding Miter Angle  = arctan( sin ( 45 ) ÷ tan( 36.8699 ))= 43.3138542102836°

Crown Molding Saw Blade Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Crown Molding Saw Blade Bevel Angle  = arcsin( cos ( 45 )  *  cos( 36.8699 )) = 34.4499007767502°

Rake Crown Molding Vertical Plane Angle = 90°
Rake Crown Spring Angle = 45°
Rake Crown Miter Angle = 35.2643896827547°
Rake Crown Bevel Angle = 30°

Hip Rafter Diamond Post Compound Angle = 25.6410063600364°
Hip Rafter Diamond Post Dihedral Angle = 34.1695442947226°      
 

 Hawkindale
 S = 45.0000°
 D = 36.8699°
 R1 = 30.9638°
 R2 = 28.0725°
 R3 = 37.8636°
 R4 = 48.8257°
 R5 = 25.6410°
 R6 = 78.8757°
 R7 = 39.2962°
 P1 = 43.3139°
 P2 = 43.3139°
 P3 = 26.5254°
 P4 = 26.5254°
 P5 = 27.9384°
 P6 = 25.2011°
 C1 = 34.4499°
 C2 = 22.8337°
 C3 = 34.4499°
 C4 = 43.3139°
 C5 = 34.4499°
 A5 = 24.3052°
 A7 = 38.6598°
 A8 = 16.7885°
 A9 = 18.7478°

Tetrahedron Angles for 9/12 irregular hip roof

D Angle = 53.1301
A Angle = 36.8699
C Angle = 30.9638
E Angle = 30.9638
B Angle = 21.1002

90-D Angle = 36.8699
90-A Angle = 53.1301
90-C Angle = 59.0362
90-E Angle = 59.0362
90-B Angle = 68.8998

The dihedral angle is the angle measured between two planes.
Dihedral Angle = 137.7996

Polygonal Pyramid Sides = 4.88
Pyramid Slant Angle = 36.8699
Pitch Angle = 36.8699
Plan Angle  = 53.1301
Working Angle = 36.8852459016393

Hip Rafter Pitch Angle = arctan( tan( Pitch Angle ) * sin( Plan Angle ))
Hip Rafter Pitch Angle = arctan( tan( 36.8699 ) * sin( 53.1301 )) = 30.9637579168711°

Hip Rafter Backing Angle = arctan( sin( Hip Rafter Pitch Angle) ÷ tan( Plan Angle ) ) 
Hip Rafter Backing Angle = arctan( sin( 30.9637579168711) ÷ tan( 53.1301 ) ) = 21.1001984464996°

Hip Rafter Plumb Backing Angle = arctan( tan( Pitch Angle ) * cos( Plan Angle ))
Hip Rafter Plumb Backing Angle = arctan( tan( 36.8699 ) * cos( 53.1301 )) = 24.2277483279458°

Hip Rafter Side Cut Angle  = arctan( cos( Hip Rafter Pitch Angle ) ÷ tan( Plan Angle ))
Hip Rafter Side Cut Angle = arctan( cos( 30.9637579168711 ) ÷ tan( 53.1301 )) = 32.7458710613747°

Jack Rafter Side Cut = arctan( cos( Pitch Angle ) ÷ tan( Plan Angle ))
Jack Rafter Side Cut = arctan( cos( 36.8699 ) ÷ tan( 53.1301 )) = 30.9637579168711°

Roof Sheathing Angle = arccos( cos( Plan Angle ) * cos( Hip Rafter Pitch Angle ))
Roof Sheathing Angle = arccos( cos( 53.1301 ) * cos( 30.9637579168711 )) = 59.0362420831289°

Frieze Block Saw Miter Angle = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Frieze Block Saw Miter Angle = arctan( sin ( 36.8699 ) ÷ tan( 53.1301 )) = 24.2277483279458°

Frieze Block Saw Blade Bevel Angle = arctan( sin( Frieze Block Saw Miter Angle ) ÷ tan( Pitch Angle ))
Frieze Block Saw Blade Bevel Angle = arctan( sin( 24.2277483279458 ) ÷ tan( 36.8699 )) = 28.6854027655007°

Square Tail Fascia Miter Angle  = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Square Tail Fascia Miter Angle  = arctan( sin ( 36.8699 ) ÷ tan( 53.1301 )) = 24.2277483279458°

Square Tail Fascia Saw Blade Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Square Tail Fascia Saw Blade Bevel Angle  = arcsin( cos ( 36.8699 )  *  cos( 53.1301 )) = 28.6854027655007°

Hip Rafter Square Tail Miter Angle  = arctan (( cos( Pitch Angle) * sin( Pitch Angle ) * cos(Plan Angle)) ÷ tan( Plan Angle))
Hip Rafter Square Tail Miter Angle  = arctan (( cos( 36.8699) * sin( 36.8699 ) * cos( 53.1301 )) ÷ tan(53.1301)) = 12.188635145255°

Hip Rafter Square Tail Saw Blade Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Hip Rafter Square Tail Saw Blade Bevel Angle  = arcsin( cos ( 36.8699 )  *  cos( 53.1301 )) = 28.6854027655007°

90° - Hip Rafter Square Tail Side Cut Angle (Foot)
R3 equals P2 rotated by C5.
R3 equals Jack Rafter Side Cut Angle rotated by the Hip Rafter Backing Angle
R3 = arctan( cos ( Hip Rafter Backing Angle ) * tan( Jack Rafter Side Cut Angle ))
R3 = arctan( cos ( 21.1001984464996 ) * tan( 30.9637579168711 )) = 29.2388545837033°

Dormer Square Tail Fascia Miter Angle  = arctan( cos ( Dormer Pitch Angle ) ÷ tan( Main Roof Pitch Angle ))
Dormer Square Tail Fascia Miter Angle  = arctan( cos ( 36.8699 ) ÷ tan( 36.8699 )) = 46.8476069378739°
Dormer Square Tail Fascia Miter Angle on Fascia = 90° - 46.8476069378739° = 43.1523930621261°

Dormer Square Tail Saw Blade Bevel Angle  = arcsin( sin ( Dormer Pitch Angle ) *  cos( Main Roof Pitch Angle ))
Dormer Square Tail Saw Blade Bevel Angle  = arcsin( sin ( 36.8699 )  *  cos( 36.8699 )) = 28.6854027655007°


Rake Stud Saw Miter Angle = Hip Rafter Plumb Backing Angle = 24.2277483279458°
Rake Stud Saw Blade Bevel Angle = arctan(cos(Hip Rafter Plumb Backing Angle)* tan(Hip Rafter Pitch Angle))
Rake Stud Saw Blade Bevel Angle = atan(cos(24.2277483279458)* tan(30.9637579168711)) = 28.6854027655007°

Rake Plate Miter Angle = Jack Rafter Side Cut Angle = 30.9637579168711°
Rake Plate Saw Blade Bevel Angle = Roof Pitch Angle = 36.8699°


Purlin Miter Angle  = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Purlin Miter Angle  = arctan( sin ( 36.8699 ) ÷ tan( 53.1301 ))= 24.2277483279458°

Purlin Saw Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Purlin Saw Bevel Angle  = arcsin( cos ( 36.8699 )  *  cos( 53.1301 )) = 28.6854027655007°

Crown Molding Miter Angle  = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Crown Molding Miter Angle  = arctan( sin ( 36.8699 ) ÷ tan( 53.1301 ))= 24.2277483279458°

Crown Molding Saw Blade Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Crown Molding Saw Blade Bevel Angle  = arcsin( cos ( 36.8699 )  *  cos( 53.1301 )) = 28.6854027655007°

Rake Crown Molding Vertical Plane Angle = 106.2602°
Rake Crown Spring Angle = 36.8699°
Rake Crown Miter Angle = 30.9637579168711°
Rake Crown Bevel Angle = 21.1001984464996°

Hip Rafter Diamond Post Compound Angle = 19.7988783552241°
Hip Rafter Diamond Post Dihedral Angle = 26.9814151018306°      
 


 Hawkindale
 S = 36.8699°
 D = 53.1301°
 R1 = 30.9638°
 R2 = 12.1886°
 R3 = 29.2389°
 R4 = 32.7459°
 R5 = 19.7989°
 R6 = -70.4746°
 R7 = 20.1197°
 P1 = 24.2277°
 P2 = 30.9638°
 P3 = 18.3085°
 P4 = 26.5174°
 P5 = 46.8476°
 P6 = 11.2289°
 C1 = 28.6854°
 C2 = 10.6739°
 C3 = 21.1002°
 C4 = 30.9638°
 C5 = 21.1002°
 A5 = 17.9808°
 A7 = 24.2277°
 A8 = 4.4464°
 A9 = 12.1886°

Ad Quadratum - Seed of Life - Ad Triangulum Router Templates

I made some router templates for the geometric symbols Ad Quadratum - Seed of Life - Ad Triangulum. These router templates were used for the window trim and soffit  knee braces. 

The router templates weren't perfect enough so I had some custom router templates made out of 1/2" HDPE by www.routertemplate.net. Well it turns out that www.routertemplate.net is owned by Brenton Green who also owns www.digiconsoo.com. Digital Construction Solutions does CNC fabrication of residential and light commercial construction products.   Groin vaults, eyebrow and barrel roofs and a lot of other really neat geometrical construction related products. Check out his websites.

This picture is the base template, 18" square with a 12" square cut out.  

It's pretty easy to align with the centering marks.  

After the base is aligned with the centering marks the circle template is inserted into the base. The circle template is used for all three symbols.  The templates are setup for a 60° V-Groove router bit with a 5/8" guide bushing. This template has an 3 5/8" diameter hole to allow for the 3" circle.

Here's I'm routing the seed of life(daisy wheel). The template has 1/4" holes spaced around the equilateral triangle for a 1/4" dowel. The router base also has a 1/4" hole drilled into the router base to swing an 1 1/2" radius for the daisy wheel arcs.  

Route the three arcs of the daisy wheel and then reverse the template to route in the other 3 arcs of the daisy wheel.

Here's the 4x4 redwood window header with 3 daisy wheels routed.

Here I was testing the custom router templates for the geometric symbols Ad Quadratum - Seed of Life - Ad Triangulum.

Picture of the knee brace and window trim routed.

Knee brace and window trim installed, except for the window header. 

Some pictures of the finished product.

I tried using a diamond tip router bit to route the granite, but as you can see the diamond tip router bit didn't really work. After routing the Ad Triangulum into the granite the diamond tip router bit was diamond-less even through I used diamond coolant with the router bit.

Looks like the granite is too tough to route with the diamond tip router bit I had. ($65.00)

Hexagon and Octagon Roof Framing Geometry

Tait Chappell told me about the advance timber framing class at Fox Maple they put on last week where the students built a hexagon 13.5/12 common pitch with two dormers at 22/12 valleys joining into rotated purlin headers. So I decided to draw out a similar hexagon plan. I couldn't let Tait have all the fun.

The day I started to draw out  the hexagon roof plan my son Brian told me a funny, but sad story that needs addressing. He's been working for a company this last month that builds commercial buildings, about 25,000 sqft a year for the last 10 years. So they know what their doing, right??? We'll their current job, a prevailing wage job at $62.00 an hour, was bid too low so the owner of the company decided that he and his partner would cut and stack the octagon roof. Well, the owner cut the octagon hips as if it were a normal hip rafter and of course the hip rafters didn't fit. Then my son Brian explained to them that the octagon hip rafter rafters are not at the common rafter pitch and their not at the 90° eave angle hip rafter slope. He told them how to calculate the octagon hip rafters the old fashion way. Use the rise and the hip rafter run to calculate the octagon hip rafter slope angle. 

So both of them had Construction Master Calculators. So why is it that they couldn't figure out the octagon hip rafter slope angle? Their ignorance? Or the Construction Master Calculator? It's 2012, why in the world world anyone call their calculator a "Master" when it can't even calculate an octagon hip rafter that the Romans and Greeks did 2,000+ years ago. Or the French carpenters did  500+ years ago.

Here's a drawing showing how to calculate the hip rafter slope angle for an octagon roof the old fashion way using a CMC calculator. Or they could have purchased a Chappell Master Framing Square that has the Hexagon and Octagon roof framing information printed on the square.

Using the Chappell Master Framing Square, line 1 under 8:12, .6159 . Framing Square usage: 6.159 on the tongue and 10 on the body to draw out the plumb and level cuts of the octagon hip rafter. 

Or arctan(.6159) = 31.62892°

Using the Chappell Master Framing Square, line 2 under 8:12, 2.902 * 56" = 162.4854", it doesn't get any easier than that.

Using the Chappell Master Framing Square, line 3 under 8:12, 3.069 * 56" = 171.8648", it doesn't get any easier than that.

Here's the drawing I started working on. Hexagon roof with13.5: common rafter pitch and 22:12 dormers into a rotated purlin header. The hexagon was drawn out using Ad Triangulum ( equilateral triangle rotated within the circle). Just like the Romans and Greeks did 2,000+ years ago. Do the French carpenters of today use Ad Triangulum to draw out hexagons?

Rafter Tools is an android app with functions to calculate complex roof framing angles including Hexagon and Octagon roofs.