Developmental Geometry for Polygon Roof Framing

S   is the length of any side of the polygon
N   is the number of sides
R   is the Radius of the circumscribed circle
apothem  is the Radius of the inscribed circle
pi is PI, approximately 3.14159

central angle = 360 ÷ N
working angle = 360 ÷ (N × 2)
plan angle = (180 - central angle) ÷ 2
apothem multiplier = secant(360° ÷ ( N × 2 ))
side wall multiplier = tan((360° ÷ (N × 2 )) × 2
projection angle a = 360 ÷ N
projection angle b = 90 - projection angle a 
miter angle = 12.0 * tan( working angle)
hip run per foot = 12.0  / cos( working angle)
adjacent run = 12 * tan(working angle)
unit rise = 12 * tan(pitch angle)
CMC irregular pitch angle = arctan( unit rise ÷ adjacent)

working angle multipler  = tan(360 ÷ (N × 2))× 2
rafter multipler = 1 ÷ cos pitch angle
hip multipler = 1 ÷ cos hip angle
rise multipler = 1 ÷ tan pitch angle

inscribed radius = apothem
inscribed radius = S / (2 * tan(360 ÷ N))
inscribed radius = R * cos(360 ÷ N)

Common Rafter Run = apothem
Hip Rafter Run = Radius
Hip Rafter Run = Common Rafter Run × apothem multipler
Common Rafter Run = S ÷ working angle multipler
S = Common Rafter Run × working angle multipler
Common Rafter Length = Common Rafter Run × rafter multipler
Hip Rafter Length = Hip Rafter Run × hip multipler
Common Rafter Rise = Common Rafter Run × rise multipler

Common Rafter Span =  Common Rafter Run × 2
Common Rafter Length = Common Rafter Run ÷ cos(Pitch Angle)
Common Rafter Rise = Common Rafter Run × tan(Pitch Angle)

Hip Rafter Angle = arctan(Common Rafter Rise ÷ Hip Rafter Run)
Hip Rafter Length = Hip Rafter Run ÷ cos(Hip Rafter Pitch)
                      
The cotangent cot(x) is the multiplicative inverse of tan(x), 1/tan(x).
The cosecant csc(x) is the multiplicative inverse of sin(x),... 1/sin(x).
The secant sec(x) is the multiplicative inverse of cos(x), ... 1/cos(x).

Line C-N = Polygon Side Wall Length
Line A-B = Common Rafter Run 90° to C-N
Line A-E = Common Rafter Rise 90° to A-B
Line E-B = Common Rafter Length
Line A-C = Polygon Hip Rafter Run
Line A-F = Common Rafter Rise 90° to A-C
Line F-C = Polygon Hip Rafter Length
Line B-D = Common Rafter Length 90° to C-N
Line C-D = Polygon Hip Rafter Length
Line N-D = Polygon Hip Rafter Length
Line C-J = Polygon Hip Rafter Side Cut Rise 90° to A-C Intersects Line B-D
Line H-C = Polygon Hip Rafter Side Cut Rise 90° to F-C
Line F-H = Hip Side Cut Angle Line At Ridge

Line A-G = Hip Backing Rise 90° to F-C Also Altitude Of Hip Length Line F-C  (Backing Angle Rise)
Line A-K = Hip Backing Rise At Same Angle As Plan Angle (Backing Angle Rise)
Line A-M = 90° to K-A And Intersects Eave Line N-M (Backing Angle Run)
Line K-M = Intersects Eave Line N-M


Line F-A ÷ Line C-A = Hip Pitch Angle
Line A-B ÷ Line C-B = Plan Angle
Line C-B ÷ Line A-B = Polygon Working Angle
Line B-D ÷ Line C-B = Sheathing Angle
Line D-B ÷ Line C-B = Jack Rafter Side Cut Angle
Line C-B ÷ Line D-B = Jack Rafter Side Cut Angle on Face Of The Plumb Cut
Line K-A ÷ Line M-A = Hip Backing Angle
Line H-C ÷ Line C-F = Hip Side Cut Angle At Ridge