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 CN = Polygon Side Wall Length Line AB = Common Rafter Run 90° to CN Line AE = Common Rafter Rise 90° to AB Line EB = Common Rafter Length Line AC = Polygon Hip Rafter Run Line AF = Common Rafter Rise 90° to AC Line FC = Polygon Hip Rafter Length Line BD = Common Rafter Length 90° to CN Line CD = Polygon Hip Rafter Length Line ND = Polygon Hip Rafter Length Line CJ = Polygon Hip Rafter Side Cut Rise 90° to AC Intersects Line BD Line HC = Polygon Hip Rafter Side Cut Rise 90° to FC Line FH = Hip Side Cut Angle Line At Ridge Line AG = Hip Backing Rise 90° to FC Also Altitude Of Hip Length Line FC (Backing Angle Rise) Line AK = Hip Backing Rise At Same Angle As Plan Angle (Backing Angle Rise) Line AM = 90° to KA And Intersects Eave Line NM (Backing Angle Run) Line KM = Intersects Eave Line NM Line FA ÷ Line CA = Hip Pitch Angle Line AB ÷ Line CB = Plan Angle Line CB ÷ Line AB = Polygon Working Angle Line BD ÷ Line CB = Sheathing Angle Line DB ÷ Line CB = Jack Rafter Side Cut Angle Line CB ÷ Line DB = Jack Rafter Side Cut Angle on Face Of The Plumb Cut Line KA ÷ Line MA = Hip Backing Angle Line HC ÷ Line CF = Hip Side Cut Angle At Ridge
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WindsorOne Protected Trim
 SBE Builders 5305 Laguna Ct. Discovery Bay, California 94505 (925) 6346022 • Fax: (925) 6346022
Commercial & Residential Framing Contractor American Carpenter and Builder California Licensed General Building Contractor CA LIC.# 546126 
Roof Framing Geometry

Rafter Tools for Android
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Crown Molding Tools
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Canadian & American Geometric Roof Framing Development
using a Steel Framing Square
Geometric Roof Framing Development for Irregular Hip Roof Calculations
using a Steel Framing Square
Geometric Development of Tetrahedron
using a Steel Framing Square
Hip Rafter Backing Bevel Angles from 1565 to 2013