# Rhombicosidodecahedron

In geometry, the rhombicosidodecahedron, or small rhombicosidodecahedron,
is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.

# Archimedean solid

In geometry, an Archimedean solid is a highly symmetric, semi-regular
convex polyhedron composed of two or more types of regular polygons meeting
in identical vertices. They are distinct from the Platonic solids,
which are composed of only one type of polygon meeting in identical vertices,
and from the Johnson solids, whose regular polygonal faces do not meet in identical vertices.

Rhombicosidodecahedron Vertex - Deck -- Ground Plan.

Rhombicosidodecahedron Roof Slope Angles with hip rafter offset for roof plane alignment.

Rhombicosidodecahedron Roof Surface Unfolded.

Rhombicosidodecahedron Roof Surface Angles.
Cut Square, Triangle & Pentagon pieces,
then miter sides of Square, Triangle and Pentagon using Hip Rafter Backing Angle.
The same as cutting roof sheathing. The hip rafters will be different sizes and the hip rafters will not be centered on the hip rafter run line.

Rhombicosidodecahedron Roof  Sheathing with a SkillSaw
1. Cut 2 half square triangles with two legs at 12", the length of the hip rafter and a base leg of 16.97".  45° roof sheathing angles and 45° jack rafter side cut angles on the roof surface. Hip Rafter Backing Angle = Saw Blade bevel Angle = 18.46°
2. Cut 1 equilateral triangle with edge lengths of 12". 60° roof sheathing angles and 30° jack rafter side cut angles on the roof surface. Hip Rafter Backing Angle = Saw Blade bevel Angle = 14.98°
3. Cut 1 triangle for the pentagon side piece. Two legs at 12" and a base leg of 19 7/16". 36° roof sheathing angles and 54° jack rafter side cut angles on the roof surface. Hip Rafter Backing Angle = Saw Blade bevel Angle = 22.39°

Rhombicosidodecahedron Hip Rafters with equal length miter line plane.
```Hip Rafter Miter Angles
Hip Rafter Width = 1.00000

Roof Eave Angle for Hip Rafter A = 102.62142
Hip Rafter A Slope Angle = 12.93932
Roof Slope Angle A = 14.98460
Roof Slope Angle B = 18.46161
Plan Angle on Side A = 59.13432
Plan Angle on Side B = 43.48710

Roof Eave Angle for Hip Rafter B = 77.37858
Hip Rafter B Slope Angle = 12.93932
Roof Slope Angle B = 18.46161
Roof Slope Angle C = 22.39277
Plan Angle on Side B = 43.48708
Plan Angle on Side C = 33.89150

Plan Miter Peak Angle = 93.02582
Hip Rafter A Offset Perpendicular = 0.36180
Hip Rafter B Offset Perpendicular = 0.58541

Hip Rafter A Miter Line Peak Plan Angle 1 = 32.53731
Hip Rafter A Miter Line Peak Plan Angle 2 = 57.46269

Hip Rafter B Miter Line Peak Plan Angle 1 = 60.48851
Hip Rafter B Miter Line Peak Plan Angle 2 = 29.51149

Hip Rafter A Miter Line Slope Angle R5P = 10.96195

Hip Rafter B Miter Line Slope Angle R5P = 6.45702

Hip Rafter A R5B = 9.46376

Hip Rafter A R5B = 9.46376

Hip Rafter A - R4P Angle = 31.87278

Hip Rafter B - R4P Angle = 59.85263

Hip Rafter A
Compound Miter Angle and Saw Blade Bevel Angle
Settings for Cutting Material Laying Flat-- Side Face
Hip Rafter A Miter Angle = 12.93932
Hip Rafter A Saw Blade Bevel Angle = 57.46269

Hip Rafter B
Compound Miter Angle and Saw Blade Bevel Angle
Settings for Cutting Material Laying Flat-- Side Face
Hip Rafter B Miter Angle = 12.93932
Hip Rafter B Saw Blade Bevel Angle = 29.51149

Hip Rafter A
Compound Miter Angle and Saw Blade Bevel Angle
Settings for Cutting Material On Top Edge

Hip Rafter A Tetrahedron Angles
D Angle = 77.06068
A Angle = 32.53731
C Angle = 31.87278
E Angle = 10.96195
B Angle = 6.91718
90-D Angle = 12.93932
90-A Angle = 57.46269
90-C Angle = 58.12722
90-E Angle = 79.03805
90-B Angle = 83.08282

Saw Miter Angle = 58.12722
Saw Blade Bevel Angle = 6.91718
Bevel Angle (Compound Angle) = 12.93932

Hip Rafter B
Compound Miter Angle and Saw Blade Bevel Angle
Settings for Cutting Material On Top Edge

Hip Rafter B Tetrahedron Angles
D Angle = 77.06068
A Angle = 60.48851
C Angle = 59.85263
E Angle = 6.45702
B Angle = 11.23695
90-D Angle = 12.93932
90-A Angle = 29.51149
90-C Angle = 30.14737
90-E Angle = 83.54298
90-B Angle = 78.76305

Saw Miter Angle = 30.14737
Saw Blade Bevel Angle = 11.23695
Bevel Angle (Compound Angle) = 12.93932```

Rhombicosidodecahedron Dihedral Angles and Hip Rafter Backing Angles

Angular Dimension (159.09484°)
Dihedral Angle = 159.094842552°
30° - 7.6226318595° ... Hip Rafter Backing Angle
45° - 13.2825255889°... Hip Rafter Backing Angle

Face Angle A = 45.00000 ... Roof Sheathing Angle
Face Angle B = 60.00000 ... Roof Sheathing Angle
Face Angle C = 102.62142 ... Eave Angle

Dihedral Angle A = 14.98459 ...Common/Profile Rafter Slope Angle
Dihedral Angle B = 18.46160...Common/Profile Rafter Slope Angle
Dihedral Angle C = 159.09486

Angular Dimension (148.28252°)
Dihedral Angle = 148.282525587°
45° - 13.2825255889°... Hip Rafter Backing Angle
54° - 18.4349488239°... Hip Rafter Backing Angle

Face Angle A = 45.00000...Roof Sheathing Angle
Face Angle B = 36.00000... Roof Sheathing Angle
Face Angle C = 77.37858... Eave Angle

Dihedral Angle A = 22.39278...Common/Profile Rafter Slope Angle
Dihedral Angle B = 18.46162...Common/Profile Rafter Slope Angle
Dihedral Angle C = 148.28251

Dihedral Angle Formulas
Dihedral angle C = arccos((cos(c) - cos(a) × cos(b)) ÷ (sin(a) × sin(b)))
Dihedral angle A = arccos((cos(a) - cos(b) × cos(c)) ÷ (sin(c) × sin(b)))
Dihedral angle B = arccos((cos(b) - cos(a) × cos(c)) ÷ (sin(a) × sin(c)))

```Tetrahedron Angles

D Angle = 59.1343387352
A Angle = 14.9845972120
C Angle = 12.9393184374
E Angle = 30.0000000000
B Angle = 7.6226318595

90-D Angle = 30.8656612648
90-A Angle = 75.0154027880
90-C Angle = 77.0606815626
90-E Angle = 60.0000000000
90-B Angle = 82.3773681405

Platonic Solid or Archimedeans Solid Angles for Compound Miter Saw Settings
Platonic Solid or Archimedeans  Solid Face = Equilateral Triangle
Jack Rafter Side Cut Angle = 30.0000000000°

Hip Rafter Pitch Angle = arctan( tan( Pitch Angle ) * sin( Plan Angle ))
Hip Rafter Pitch Angle = arctan( tan( 14.9845972120° ) * sin( 59.1343387352° )) = 12.9393184374°

Hip Rafter Backing Angle = arctan( sin( Hip Pitch Angle) ÷ tan( Plan Angle ) )
Hip Rafter Backing Angle = arctan( sin( 12.9393184374°) ÷ tan( 59.1343387352° ) ) = 7.6226318595°

Hip Rafter Side Cut Angle = arctan( cos( Hip Rafter Pitch Angle ) ÷ tan( Plan Angle ))
Hip Rafter Side Cut Angle = arctan( cos( 12.9393184374° ) ÷ tan( 59.1343387352° )) = 30.2207003332°

Dihedral Angle =  90° - 7.6226318595° - 7.6226318595° = 164.7547362810°

Settings for Cutting Hip Rafter Material On Side of material
Saw Miter Angle = 12.9393184374°
Saw Blade Bevel Angle = 59.1343387352°

Settings for Cutting Hip Rafter Material On Top Edge of material
Top Edge of Hip Rafter Saw Miter Angle = 59.7792996668°
Top Edge of Hip Rafter Saw Blade Bevel Angle = 6.5964995205°```

```Tetrahedron Angles

D Angle = 43.4870777454
A Angle = 18.4616148311
C Angle = 12.9393184374
E Angle = 45.0000000000
B Angle = 13.2825255889

90-D Angle = 46.5129222546
90-A Angle = 71.5383851689
90-C Angle = 77.0606815626
90-E Angle = 45.0000000000
90-B Angle = 76.7174744111

Platonic Solid or Archimedeans Solid Angles for Compound Miter Saw Settings
Platonic Solid or Archimedeans  Solid Face = Square
Jack Rafter Side Cut Angle = 45.0000000000°

Hip Rafter Pitch Angle = arctan( tan( Pitch Angle ) * sin( Plan Angle ))
Hip Rafter Pitch Angle = arctan( tan( 18.4616148311° ) * sin( 43.4870777454° )) = 12.9393184374°

Hip Rafter Backing Angle = arctan( sin( Hip Pitch Angle) ÷ tan( Plan Angle ) )
Hip Rafter Backing Angle = arctan( sin( 12.9393184374°) ÷ tan( 43.4870777454° ) ) = 13.2825255889°

Hip Rafter Side Cut Angle = arctan( cos( Hip Rafter Pitch Angle ) ÷ tan( Plan Angle ))
Hip Rafter Side Cut Angle = arctan( cos( 12.9393184374° ) ÷ tan( 43.4870777454° )) = 45.7767018361°

Dihedral Angle =  90° - 13.2825255889° - 13.2825255889° = 153.4349488222°

Settings for Cutting Hip Rafter Material On Side of material
Saw Miter Angle = 12.9393184374°
Saw Blade Bevel Angle = 43.4870777454°

Settings for Cutting Hip Rafter Material On Top Edge of material
Top Edge of Hip Rafter Saw Miter Angle = 44.2232981639°
Top Edge of Hip Rafter Saw Blade Bevel Angle = 9.3497035428°

Tetrahedron Angles

D Angle = 33.8915057735
A Angle = 22.3927709301
C Angle = 12.9393184378
E Angle = 54.0000000000
B Angle = 18.4349488239

90-D Angle = 56.1084942265
90-A Angle = 67.6072290699
90-C Angle = 77.0606815622
90-E Angle = 36.0000000000
90-B Angle = 71.5650511761

Platonic Solid or Archimedeans Solid Angles for Compound Miter Saw Settings
Platonic Solid or Archimedeans  Solid Face = Pentagon
Jack Rafter Side Cut Angle = 54.0000000000°

Hip Rafter Pitch Angle = arctan( tan( Pitch Angle ) * sin( Plan Angle ))
Hip Rafter Pitch Angle = arctan( tan( 22.3927709301° ) * sin( 33.8915057735° )) = 12.9393184378°

Hip Rafter Backing Angle = arctan( sin( Hip Pitch Angle) ÷ tan( Plan Angle ) )
Hip Rafter Backing Angle = arctan( sin( 12.9393184378°) ÷ tan( 33.8915057735° ) ) = 18.4349488239°

Hip Rafter Side Cut Angle = arctan( cos( Hip Rafter Pitch Angle ) ÷ tan( Plan Angle ))
Hip Rafter Side Cut Angle = arctan( cos( 12.9393184378° ) ÷ tan( 33.8915057735° )) = 55.4231060666°

Dihedral Angle =  90° - 18.4349488239° - 18.4349488239° = 143.1301023522°

Settings for Cutting Hip Rafter Material On Side of material
Saw Miter Angle = 12.9393184378°
Saw Blade Bevel Angle = 33.8915057735°

Settings for Cutting Hip Rafter Material On Top Edge of material
Top Edge of Hip Rafter Saw Miter Angle = 34.5768939334°
Top Edge of Hip Rafter Saw Blade Bevel Angle = 10.7120937647°

```