Sloped Ridge Lines
Roof with Star Footprint

Hip Rafter Angles
 SS : Common Rafter Pitch Angle = 39.805571° DD : Plan Angle = arcsin (3 ÷ 10) = 17.457603° W : Plan Angle between Eaves = 2 × 17.457603° = 34.915206° R1 : Hip Pitch Angle = 14.036243° 90 – P2 : Sheathing Angle = 22.262638° C5 : Backing Angle = 37.639917°

Valley Rafter Angles
 SS : Common Rafter Pitch Angle = 39.805571° W : Plan Angle between Eaves = 180° – (108° – 2 × 17.457603°) = 106.915206° DD : Plan Angle = 106.915206° ÷ 2 = 53.457603° R1 : Valley Pitch Angle = 33.802852° 90 + P2 : Supplement of Sheathing Angle = 119.654332° C5 : Backing Angle = 22.406606°

Angles at the Roof Peak
 First we create a Reference Plane lying on the Hip rafter ridge lines and defined by the points at the roof peak and two adjacent vertices of the pentagon. This Reference Plane forms the surface of a pentagonal pyramid 3 high. The edges of the base are 2 × 12 × cos 54° = 14.106846 long. W : Plan Angle at Vertex of Pentagon = 108° DD : Plan Angle = 108° ÷ 2 = 54° Reference Plane Run = 12 × sin 54° = 9.708204 Reference Plane Rise = 3 SS : Reference Plane Pitch Angle = arctan (3 ÷ 9.708204) = 17.172038° R1 : Hip Pitch Angle = 14.036243° 90 – P2 : Sheathing Angle = 55.233354° C5 : Backing Angle = 9.993635°

 The angle on the Reference Plane at the peak of the roof, which will substitute for W , equals 180° – (2 × 55.233354°) = 69.533292° DD : Plan Angle = 69.533292° ÷ 2 = 34.766665° The "Pitch Angle", SS , with respect to the Reference Plane is the difference between the Backing Angle returned by the Hip Rafter calculation and the Backing Angle with respect to the Reference Plane : 37.639917° – 9.993635° = 27.646282° R1 : Valley Pitch Angle = 16.630814° 90 – P2 : Sheathing Angle = 38.083030° C5 : Backing Angle = 22.406606°

Do the results make sense?
 The sum of the Sheathing Angles on the planes lying on the surface of the roof = 22.262638° + 119.654332° + 38.083030° = 180° The Backing Angle returned by the Valley Rafter calculation equals the Backing Angle returned by the calculation with respect to the Reference Plane ... 22.406606° The sum of the Reference Plane Pitch Angle with respect to level and the Valley Pitch Angle with respect to the Reference Plane equals the Valley Pitch Angle with respect to level: 17.172038° + 16.630814° = 33.802852°