Hopper Miter Angle and Saw Blade Bevel Angle Settings

Hopper angles are the same angles that are use for cutting roof sheathing. The hopper miter angle is the same as the jack rafter side cut angle and the hopper saw blade bevel angle is the same as the hip rafter backing angle.
Hopper Miter Angle = arctan (cos(Roof Pitch Angle) ÷ tan (Plan Angle))
Hopper Saw Blade Bevel Angle for Mitered Hopper = arctan( sin ( Hip Rafter Angle ) ÷ tan ( Plan Angle ))
Hopper Saw Blade Bevel Angle for Butt Joint Hopper = arctan( 90° - (sin ( Hip Rafter Angle ) ÷ tan ( Plan Angle )*2))
See the bottom of this page for more formulas for the hopper miter and bevel angles.

The first column in the table is the corner angle in degrees. The second column is the miter angle for cutting the hopper material laying flat in the compound miter saw. The third column is the bevel angle for cutting the hopper material laying flat in the compound miter saw. The fourth column is the Hip Rafter angle for the referenced wall angle. The Hip Rafter Angle is for reference only. The fifth column is the miter for a butt joint hopper and the sixth column is the bevel angle for the butt joint hopper.

Hopper Slope Angle = 33.69 °

• Common Corner Angles for mitered hopper with a slope angle of 33.69 °
• [Corner Angle] Miter : Bevel
• [rectangle] 39.76 : 23.09 [4 sided polygon] : 45° Polygon Miter Angle : Wall Angle 90° : Plan Angle 45°
• [pentagon] 31.15 : 19.03 [5 sided polygon] : 36° Polygon Miter Angle : Wall Angle 108° : Plan Angle 54°
• [hexagon] 25.66 : 16.10 [6 sided polygon] : 30° Polygon Miter Angle : Wall Angle 120° : Plan Angle 60°
• [heptagon] 21.84 : 13.93 [7 sided polygon] : 25.71° Polygon Miter Angle : Wall Angle 128.5714° : Plan Angle 64.2857°
• [octagon] 19.02 : 12.26 [8 sided polygon] : 22.5° Polygon Miter Angle : Wall Angle 135° : Plan Angle 67.5°
• [nonagon] 16.85 : 10.94 [9 sided polygon] : 20° Polygon Miter Angle : Wall Angle 140° : Plan Angle 70°
• [decagon] 15.13 : 9.87 [10 sided polygon] : 18° Polygon Miter Angle : Wall Angle 144° : Plan Angle 72°
• [dodecagon] 12.57 : 8.25 [12 sided polygon] : 15° Polygon Miter Angle : Wall Angle 150° : Plan Angle 75°
• [hexadecagon] 9.40 : 6.21 [16 sided polygon] : 11.25° Polygon Miter Angle : Wall Angle 157.5° : Plan Angle 78.75°
• [icosikaitetragon] 6.25 : 4.15 [24 sided polygon] : 7.5° Polygon Miter Angle : Wall Angle 165° : Plan Angle 82.5°

   
   Roof Pitch Angle = Hopper Slope Angle
   
   Use any of these formulas to find the Hopper Miter Angle (Jack Rafter Side cut Angle) for Mitered or Butt Joint Hoppers
Hopper Miter Angle  = arccos( sin ( Plan Angle )  ÷   cos ( Hip Rafter Backing Angle ))
Hopper Miter Angle  = arccos( sin ( Hip Rafter Angle )  ÷   sin ( Roof Pitch Angle ))
Hopper Miter Angle  = arcsin ( cos ( Plan Angle )  ×  cos ( Hip Rafter Angle ))
Hopper Miter Angle  = arcsin ( tan ( Hip Rafter Backing Angle )  ÷   tan ( Roof Pitch Angle ))
Hopper Miter Angle  = arctan( cos  ( Roof Pitch Angle )  ÷  tan ( Plan Angle ))
Hopper Miter Angle  = arctan( sin  ( Hip Rafter Backing Angle )  ÷   tan ( Hip Rafter Angle ))

Use any of these formulas to find the Hopper Saw Blade Bevel Angle (Hip Rafter Backing Angle)for Mitered Hoppers
Hopper Saw Blade Bevel Angle  = arccos( sin ( Plan Angle )  ÷  cos ( Jack Rafter Side Cut Angle ))
Hopper Saw Blade Bevel Angle  = arccos( cos ( Roof Pitch Angle )  ÷  cos ( Hip Rafter Angle ))
Hopper Saw Blade Bevel Angle  = arcsin ( sin ( Roof Pitch Angle )  ×  cos ( Plan Angle ))
Hopper Saw Blade Bevel Angle  = arcsin ( tan ( Jack Rafter Side Cut Angle )  ×  tan ( Hip Rafter Angle ))
Hopper Saw Blade Bevel Angle  = arctan( tan  ( Roof Pitch Angle )  ×  sin ( Jack Rafter Side Cut Angle ))
Hopper Saw Blade Bevel Angle  = arctan( sin  ( Hip Rafter Angle )  ÷  tan ( Plan Angle )) 


The saw blade bevel angle for butt joint hoppers is 90° - (2 ×  Hip Rafter Backing Angle)
or  90° - (2 ×  Mitered Hopper Saw Blade Bevel Angle)


Hopper Miter Angle = arctan (cos(Roof Pitch Angle)  ÷  tan (Plan Angle)) 
Hopper Saw Blade Bevel Angle  = arctan( sin  ( Hip Rafter Angle )  ÷  tan ( Plan Angle )) 
example: 
Roof Pitch Angle = 38° 
Wall Angle = 90°
Plan Angle = 45°
Hip Rafter Angle  = arctan( tan  ( 38° )  ×  sin ( 45 )) = 28.91856°

Hopper Miter Angle = arctan (cos( 38° )  ÷  tan( 45° )) = 38.23849° 
Hopper Saw Blade Bevel Angle  = arctan( sin( 28.91856° )  ÷  tan( 45° )) = 25.80683°

or
Hopper Miter Angle = arctan (cos(Roof Pitch Angle) ÷ tan (Plan Angle)) 
Hopper Saw Blade Bevel Angle  = arccos( sin ( Plan Angle )  ÷  cos ( Hopper Miter Angle ))
Hopper Miter Angle = arctan (cos( 38° )  ÷  tan( 45° )) = 38.23849° 
Hopper Saw Blade Bevel Angle  = arccos( sin( 45° )  ÷  cos( 38.23849° )) = 25.80683°