In the case of a regular roof (equal slopes), the next right triangle to be constructed,
with one leg following the hypotenuse of the triangle of the Adjoining Plan Angle,
would define part of the angle measured between the abutting ends of the Main and Adjoining fascia.
But the geometry of the development above also requires that such a triangle follow a plumb plane.
In the case of an irregular roof (unequal slopes) the intersecting fascia at the foot of the Hip rafter create a skewed (nonplumb) line and plane:
this right triangle and hence the Square Fascia Bevel Angle cannot be constructed using the development above.
However, the Main Side Sheathing Angle, 90° – P2m, found on the side of the fascia following the surface of the roof, is a known angle.
(We can construct 90° – P2m = 63.25779° by drawing another development similar to that above,
beginning with the Main Plan Angle, DD = 59.74356°, and the Main Slope Angle, SS = 30.25644°.)
Since we know the two angles on the adjacent planes of the fascia,
we can extract the tetrahedron from the stick and construct the remaining Main Side Fascia Angles as follows ...
