Octagon Gazebo
Square Tail Hip Rafters

Octogonal Gazebo
Octogonal Gazebo
Photos courtesy of John Devries Log Homes

Octogonal Gazebo Plan View
Angles on the Planes of the Roof
Sheathing Angle
arctan ( tan 67.50000° ÷ cos 26.56505° ) = 69.67114°
Jack Rafter Side Cut Angle
arctan ( cos 26.56505° ÷ tan 67.50000° ) = 20.32886°
Square Fascia Angle on the Stick
arctan ( tan 67.50000° ÷ sin 26.56505° ) = 79.50538°

Square Tail Hip Rafter Overview
Hip Rafter Pitch Angle
arctan ( tan 26.56505° × sin 67.50000° ) = 24.79409°
Hip Rafter Side Cut Angle (Peak)
arctan ( cos 24.79409° ÷ tan 67.50000° ) = 20.60785°
Hip Square Tail Side Cut Angle (Foot)
arctan ( tan 69.67114° ÷ cos 9.85418° ) = 69.94701°
Hip Rafter Miter Angle (Foot)
arctan ( sin 9.85418° ÷ tan 69.67114° ) = 3.62799°

Square Tail Hip Rafter Sections detailing the Backing Angles
Backing Angle
arctan ( sin 24.79409° ÷ tan 67.50000° ) = 9.85418°
Plumb Backing Angle
arctan ( tan 24.79409° ÷ tan 67.50000° ) = 10.83214°

Compound Angle at Hip Rafter Peak
Compound Angle Blade Bevels
arctan ( sin 65.20591° ÷ tan 20.60785° ) = 67.50000°
arctan ( sin 20.60785° ÷ tan 65.20591° ) = 9.23485°
Angle on plumb face created by Compound Angle cut
arccos ( cos 20.60785° × cos 65.20591° ) = 66.88834°
Jack Rafter Side Cut Angle created by Backing Angle cut
arctan ( tan 20.60785° × cos 9.85418° ) = 20.32886°

Compound Angle at Square Tail Hip Rafter Foot
Compound Angle Blade Bevels
arctan ( sin 86.37201° ÷ tan 69.94701° ) = 20.01600°
arctan ( sin 69.94701° ÷ tan 86.37201° ) = 3.40858°
Angle on face created by Compound Angle cut
arccos ( cos 69.94701° × cos 86.37201° ) = 88.75674°
Sheathing Angle created by Backing Angle Cut
arctan ( tan 69.94701° × cos 9.85418° ) = 69.67114°

Tertahedra extracted form Compound Angle at Square Tail Hip Rafter Foot
Compound Angle on Square Tail Fascia
Square Tail Fascia intercepts Hip Rafter
Square Fascia Miter Angle
arctan ( sin 26.56505° ÷ tan 67.50000° ) = 10.49462°
Square Fascia Angle on the Stick
arctan ( tan 67.50000° ÷ sin 26.56505° ) = 79.50538°
Square Fascia Blade Bevel Angle
arcsin ( cos 26.56505° × cos 67.50000° ) = 20.01600°
Angle on Matching Faces of Fascia
arccos ( cos 69.67114° × cos 79.50538° ) = 86.37201°

Summary of Cutting Angles and Over 12 Values
Common Rafter Pitches = 6/12
Plan Angle between Hip Roof Eaves = 135°
Common Rafter Pitch Angle = 26.56505° ... 6"
Plan Angle at Eave = 67.50000° ... 29"
Plan Angle at Center = 22.50000° ... 5"
Hip Rafter Pitch Angle = 24.79409° ... 5 9/16"
Jack Rafter Side Cut Angle = 20.32886° ... 4 7/16"
Sheathing Angle = 69.67114° ... 32 3/8"
Backing Angle = 9.85418° ... 2 1/16"
Plumb Backing Angle = 10.83214° ... 2 5/16"
Hip Rafter Side Cut Angle (Peak) = 20.60785° ... 4 1/2"
Blade Bevel along Hip Rafter Side Cut Angle (Peak) = 9.23485° ... 1 15/16"
Hip Square Tail Side Cut Angle (Upper Shoulder at Foot) = 69.94701° ... 32 7/8"
Blade Bevel along Hip Square Tail Side Cut Angle (Foot) = 3.40858° ... 0 11/16"
Square Tail Fascia Miter Angle = 10.49462° ... 2 1/4"
Square Tail Fascia Angle on the Stick = 79.50538° ... 64 3/4"
Square Tail Fascia Blade Bevel Angle = 20.01600° ... 4 3/8"
Hip Rafter Miter Angle (on Side Face at Foot) = 3.62799° ... 0 3/4"
Blade Bevel along Hip Rafter Miter Angle = 20.01600° ... 4 3/8"

NOTES (Refer to the Supplementary Sketches below)
The Hip Rafter Side Cut Angle (Foot) and the affiliated Blade Bevel Angle produce PLUMB PLANES at the Hip Rafter Foot.
Hip Rafter Side Cut Angle (Foot) = 65.47409° ... 26 5/16"
Blade Bevel along Hip Rafter Side Cut Angle (Foot) = 22.79510° ... 5 1/16"
The Square Tail Fascia Miter and Bevel Angles are the same as required to cut a Purlin at its intercept with the Hip Rafter.

Supplementary Sketches

Compound Angle at Plumb Tail Hip Rafter Foot
Hip Rafter Side Cut Angle (Foot)
arctan ( cos 24.79409° × tan 67.50000° ) = 65.47409°
Compound Angle Blade Bevels
arctan ( sin 65.20591° ÷ tan 65.47409° ) = 22.50000°
arctan ( sin 65.47409° ÷ tan 65.20591° ) = 22.79510°
Angle on face created by Compound Angle cut
arccos ( cos 65.47409° × cos 65.20591° ) = 79.97501°
Sheathing Angle created by Backing Angle Cut
arctan ( tan 24.79409° ÷ sin 9.85418° ) = 69.67114°

Compound Angle on Purlin at intercept with Hip Rafter
Compound Angle Blade Bevels
arctan ( sin 79.50538° ÷ tan 69.67114° ) = 20.01600°
arctan ( sin 69.67114° ÷ tan 79.50538° ) = 9.85418°
Angle on face intercepting the Hip Rafter
arccos ( cos 69.67114° × cos 79.50538° ) = 86.37201°

Compound Angle on Jack Rafter at intercept with Hip Rafter
Compound Angle Blade Bevels
arctan ( sin 63.43495° ÷ tan 20.32886° ) = 67.50000°
arctan ( sin 20.32886° ÷ tan 63.43495° ) = 9.85418°
Angle on face intercepting the Hip Rafter
arccos ( cos 20.32886° × cos 63.43495° ) = 65.20591°


Joe Bartok
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