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Development of the 6 Tetrahedrons for Crown Molding Angles
Using Plane and Spherical Trigonometry

24 Sided Polygon Crown Molding Angles

Crown Molding Tools
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Crown Molding Tools  

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 ))

Crown Angle Crown Angle

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.

Crown 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 ))
Angle C = arctan( tan( 52 ) * sin( 45 )) = 42.15
Angle B = arccos( cos( A ) cos( C ))
Angle E = arctan( cos( A ) tan( D ))
Angle E = arctan( cos( 52 ) tan( 45 )) = 31.62 = Crown Miter Angle
Angle B = arccos( cos( 52 ) cos( 42.15 )) = 33.86 = Crown Bevel Angle

So we now know to use the crown slope angle for the development of the tetrahedron for horizontal plane crown miter and bevel angles.


Crown Angle

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
Hip Pitch Angle = arctan (tan( Spring Angle ) * sin( Plan Angle ))
Hip Pitch Angle = arctan (tan( 38 ) * sin( 45 )) = 28.92
Hip Backing Angle = arctan (sin( Hip Pitch Angle) tan( Plan Angle ))
Hip Backing Angle = arctan (sin( 28.92 ) tan( 45 )) = 25.81
Jack Rafter Side Cut Angle = arccos(sin( Plan Angle ) cos( Hip Backing Angle ))
Jack Rafter Side Cut Angle = arccos(sin( 45 ) cos( 25.81 )) = 38.34

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
Angle B = arccos( cos( 38 ) cos( 28.2 )) = 25.81

Crown Angle

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 = 25.81
Angle A = 38.24
Angle C = arctan( tan( A ) * sin( D ))
Angle C = arctan( tan( 38.24 ) * sin( 25.81 )) = 18.93
Angle B = arccos( cos( A ) cos( C ))
Angle E = arctan( cos( A ) tan( D ))
Angle E = arctan( cos( 38.24 ) tan( 25.81 )) = 58.38
Crown Miter Angle = 90° - 58.38° = 31.62°
Angle B = arccos( cos( 38.24 ) cos( 18.93 )) = 33.86
Crown Bevel Angle = 33.86°


Crown Angle

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
Angle C = arctan( tan( A ) * sin( D ))
Angle C = arctan( tan( 38 ) * sin( 56.31 )) = 33.03
Angle B = arccos( cos( A ) cos( C ))
Angle E = arctan( cos( A ) tan( D ))
Angle E = arctan( cos( 38 ) tan( 56.31 )) = 27.71 = Rake Crown Miter Angle
Angle B = arccos( cos( 38 ) cos( 33.03 )) = 19.97 = Rake Crown Bevel Angle

So we now know to use the crown spring angle for the development of the tetrahedron for vertical plane crown miter and bevel angles.


Crown Angle

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
Hip Pitch Angle = arctan (tan( 90 - Spring Angle ) * sin( Plan Angle ))
Hip Pitch Angle = arctan (tan( 90 - 38 ) * sin( 56.31 )) = 46.80
Hip Backing Angle = arctan (sin(Hip Pitch Angle) tan(Plan Angle))
Hip Backing Angle = arctan (sin (46.80 ) tan( 56.31 )) = 25.92
Jack Rafter Side Cut Angle = arccos(sin(Plan Angle) cos(Hip Backing Angle))
Jack Rafter Side Cut Angle = arccos(sin( 56.31 ) cos( 25.92 )) = 22.32

Angle D = 56.31
Angle A = 52.00
Angle C = arctan( tan( A ) * sin( D ))
Angle C = arctan( tan( 52 ) * sin( 56.31 )) = 46.80
Angle B = arccos( cos( A ) cos( C ))
Angle E = arctan( cos( A ) tan( D ))
Angle E = arctan( cos( 52 ) tan( 56.31 )) = 22.32
Angle B = arccos( cos( 52 ) cos( 56.31 )) = 25.92


Crown Angle

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
Angle A = 22.32
Angle C = arctan( tan( A ) * sin( D ))
Angle C = arctan( tan( 22.32 ) * sin( 25.92 )) = 10.17
Angle B = arccos( cos( A ) cos( C ))
Angle E = arctan( cos( A ) tan( D ))
Angle E = arctan( cos( 22.32 ) tan( 25.92 )) = 62.29
Crown Rake Miter angle = 90 - 62.29 = 27.71
Angle B = arccos( cos( 22.32 ) cos( 10.17 )) = 19.97
Crown Rake Bevel Angle = 19.97

 


Crown Angle
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
Crown Miter Angle = arctan (sin( Spring Angle) tan ( Wall Corner Angle 2 ))
Crown Miter Angle = arctan (sin( 38) tan( 45)) = 31.62
 
Crown Bevel Angle = arcsin (cos( Spring Angle ) * cos( Wall Corner Angle 2 ))
Crown Bevel Angle = arcsin (cos( 38) * cos( 45)) = 33.86

or you can think of the wall angle as the miter bisect angle
Crown Miter Angle = arctan (sin( Spring Angle) tan ( Wall Miter Bisect Angle ))
Crown Miter Angle = arctan (sin( 38) tan( 45)) = 31.62
 
Crown Bevel Angle = arcsin (cos( Spring Angle ) * cos( Wall Miter Bisect Angle))
Crown Bevel Angle = arcsin (cos( 38) * cos( 45)) = 33.86

or you use the crown slope angle instead of the crown spring angle for horizontal plane crown molding angles
Crown Miter Angle = arctan (cos(Crown Slope Angle) * tan( Wall Miter Bisect Angle ))
Crown Miter Angle = arctan (cos( 52) * tan( 45)) = 31.62
 
Crown Bevel Angle = arcsin (sin( Crown Slope Angle ) * sin( Wall Miter Bisect Angle))
Crown Bevel Angle = arcsin (sin( 52 ) * sin( 45)) = 33.86

Trigonometric Crown Angle Formulas for Vertical Plane Crown Molding Angles (cathedral/vaulted ceilings/rake gables)
Crown Miter Angle = arctan (cos( Spring Angle) tan((180 - (Roof Pitch Angle * 2)) 2 ))
Crown Miter Angle = arctan (cos( 38) tan((180 - (33.69 * 2)) 2)) = 27.71
Crown Miter Angle = arctan (cos( 38) tan(56.31) = 27.71
 
Crown Bevel Angle = arcsin (sin( Spring Angle ) * cos((180 - (Roof Pitch Angle * 2)) 2))
Crown Bevel Angle = arcsin (sin( 38) * cos( 56.31)) = 19.97


 
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

Crown Spring Angles

Crown Spring Angle
Polygon Crown Molding Miter Angles and Bevel Angles

12 Sided Polygon Crown Molding

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SBE Builders
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