Crown Molding Tools
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Upside down and backwards in a miter box or compound miter saw has been used
for the last 200+ years by the majority of carpenters to cut
crown molding
miter and bevel angles. Right sideup and backwards for rake crown molding
cuts were developed long before our time, but even today there's still
confusion about the correct crown miter and bevel angles when cutting rake
crown molding laying flat
in the miter box or compound miter saw.
By using plane and trigonometry for the development of the 6 tetrahedrons
for crown molding angles we are able to see how all of the different ways
of cutting crown molding are interrelated and how to mathematically
calculate the correct miter and bevel angles for rake crown molding.
An irregular tetrahedron is a polyhedron with four planar faces and six edges, that do not have equal edge lengths. Tetrahedrons have a trigonometric relationship between each triangular face of the tetrahedron. Once we know 2 angles of the tetrahedron we can use trigonometry to calculate the 6 edge lengths of the developed tetrahedron and each angle. The irregular tetrahedron is like a pyramid with an irregular length triangular base. Face D , the base of the tetrahedron kernel, is created by extracting a pyramid surface from the crown miter corner. The dihedral angle (angle between two planes) is deveopled from the tetrahedron kernel, which gives us our crown bevel angle B.
The dihedral angle between the planes in a irregular tetrahedron can be developed from the triangular faces with the law of cosines.
The dihedral angle is = 90  arccos( sin( D ) ÷ cos( E )) So Angle B = arccos( sin( D ) ÷ cos( E )) or Angle B = arccos( cos( A ) ÷ cos( C )) 
We'll start off the development of the tetrahedron with the two known angles for the typical horizontal plane 90° corner angle with a crown spring angle of 38° Angle D = 90° ÷ 2 = 45° Angle C = arctan( tan( A ) * sin( D )) Angle C = arctan( tan( 38° ) * sin( 45° )) = 28.92° Angle B = arccos( cos( A ) ÷ cos( C )) Angle E = arctan( cos( A ) ÷ tan( D )) Angle E = arctan( cos( 38° ) ÷ tan( 45° )) = 38.23° = Crown Miter Angle Angle B = arccos( cos( 38° ) ÷ cos( 28.2° )) = 25.81° = Crown Bevel Angle But wait! Every carpenter knows that crown molding with a spring angle
of 38° uses a Crown Miter Angle of 31.62° and a Crown
Bevel Angle of 33.86°. So what went wrong with the development
of the tetrahedron angles. We used the wrong angle for the surface angle A.
The correct angle to use for the surface angle A is 52°.
Which is the complement angle of the triangle A,
or the crown slope angle. So the correct angle to use for
the surface angle A is the crown slope angle and not the
crown spring angle. 

This time we'll start off the development of the tetrahedron with the two known angles for the typical 90° corner angle with a crown slope angle of 52°
Angle D = 90° ÷ 2 = 45°
Angle C = arctan( tan( A ) * sin( D )) So we now know to use the crown slope angle for the development of the tetrahedron for horizontal plane crown miter and bevel angles. 

Now we'll develop the tetrahedron for crown molding angles using roof framing angles. To develop the correct miter and bevel angles for crown molding cuts from roof framing angles we need to develop 2 tetrahedrons. The first tetrahedron will be the tetrahedron kernel which enables us to develop another tetrahedron for the crown spring angle for the development of the tetrahedron for horizontal plane crown miter and bevel angles. For the crown development of the tetrahedron we'll use the two known angles for the typical roof slope angle 33.69° with a crown spring angle of 38°. The tetrahedron kernel is developed from the corner plan angle and roof pitch angle.
Plan Angle = 90° ÷ 2 = 45° 

Now we'll use Angle B, the hip rafter backing angle, and Angle E, the jack rafter side cut angle, to develop the next tetrahedron.


Now we'll develop the tetrahedron for rake crown molding angles. To develop the correct miter and bevel angles for rake crown molding cuts we need to use the crown spring angle for the development of the tetrahedron for vertical plane crown miter and bevel angles. For the rake crown development of the tetrahedron we'll use the two known angles for the typical roof slope angle 33.69° with a crown spring angle of 38°
Angle D = (180°  ( 33.69° * 2 )) ÷ 2 = 56.31° So we now know to use the crown spring angle for the development of the tetrahedron for vertical plane crown miter and bevel angles. 

Now we'll develop the tetrahedron for rake crown molding angles using roof framing angles. To develop the correct miter and bevel angles for rake crown molding cuts from roof framing angles we need to develop 2 tetrahedrons. The first tetrahedron will be the tetrahedron kernel which enables us to develop another tetrahedron for the crown spring angle for the development of the tetrahedron for vertical plane crown miter and bevel angles. For the rake crown development of the tetrahedron we'll use the two known angles for the typical roof slope angle 33.69° with a crown spring angle of 38°. The tetrahedron kernel is developed from the corner plan angle and roof pitch angle.
Plan Angle = (180°  ( 33.69° * 2 )) ÷ 2 = 56.31° 

Now we'll use Angle B, the hip rafter backing angle, and Angle E, the jack rafter side cut angle, to develop the next tetrahedron. Angle D = 25.92°


Trigonometric Relationships between Tetrahedron Angles by Joe Bartok
Joe Bartok's Trigonometric Relationships between Tetrahedron Angles web page is the most significant treatise on roof framing angles in the last 100 years. If your a roof framer print out Joe's Trigonometric Relationships between Tetrahedron Angles web page and study the trigonometric relationships between each of the triangles. If your an interior trim carpenter, then just use the crown angle formulas below.
Trigonometric Crown Angle Formulas for Horizontal Plane Crown Molding Angles
Trigonometric Crown Angle Formulas for Vertical Plane Crown Molding Angles (cathedral/vaulted ceilings/rake gables)

Development of Tetrahedron Geometry based on Joe Bartok's Tetrahedron Developmental Geometry Development of Tetrahedron Geometry Development of Tetrahedron for Crown Molding Geometry Using Spring Angle and Wall Angle Development of Tetrahedron for Crown Molding Geometry Using Slope Angle and Corner Plane Angle Development of Tetrahedron for Crown Rake Molding Geometry 
Crown Moulding Miter Angle and Bevel Angle Settings Calculators Crown Spring Angle = 52° Crown Spring Angle = 45° Crown Spring Angle = 38° Wall Angle Settings Rake Crown Spring Angle = 38° Rake to Horizontal Crown Spring Angle = 38° Polygon 2 Cord Exterior Rake Gable Crown Molding Miter Angle and Bevel Angle Settings Polygon 2 Cord Interior Rake Gable Crown Molding Miter Angle and Bevel Angle Settings Polygon Crown Molding Angles Development & Calculator Octagon Roof Exterior Crown Molding Angles Development Calculators Octagon Roof Interior Crown Molding Angles Development Calculators Corner A Crown Molding Calculator Corner B Crown Molding Calculator Corner C Crown Molding Calculator Corner D Crown Molding Calculator 
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Roof Framing Geometry

Rafter Tools for Android
With Quad Tetrahedra Analytic Algorithm Technology.
Helping apprentice carpenters on their journey to becoming master carpenters with Certification in roof framing.
Crown Molding Tools
crownmoldingtools.com
Canadian & American Geometric Roof Framing Development
using a Steel Framing Square
Geometric Roof Framing Development for Irregular Hip Roof Calculations
using a Steel Framing Square
Geometric Development of Tetrahedron
using a Steel Framing Square
Hip Rafter Backing Bevel Angles from 1565 to 2013