Square Cut Fascia intersects foot of Hip Rafter
Developments of Compound Angles


Development and Trigonometric Proof of the Square Fascia Miter Angle, given the Reciprocal Main Slope Angle and Adjoining Side Plan Angle
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 (non-plumb) 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 ...
Tetrahedron of the Compound Angle ... Main Side Square Fascia
Development of Compound Angle ... Main Side Fascia
To solve the angles for the Adjoining Side Square Fascia we repeat the steps above to construct the angles on the adjacent faces of the stick, 90 SFMa = 67.58488 and 90 P2a = 39.52120, and given these two angles we proceed thus (same pattern as the Main Side Fascia development, different angles) ...
Development of Compound Angle ... Adjoining Side Fascia
CHECKING OUR WORK ... note that in both of the above developments the angles at the abutting ends of both the Main and Adjoining Side Fascia, SFCm and SFCa respectively, must be EQUAL (for this roof, 72.89392).


Proof ... Saw Blade Bevel along the
miter line of the Main Side Sheathing Angle
equals the Adjoining Side Backing Angle


Trigonometric Proof of the Saw Blade Bevel along the miter line of the Main Side Sheathing Angle
Using the same reasoning, we can show that the Saw Blade Bevel along the miter line of the Adjoining Side Sheathing Angle, 90 P2a, is: arctan (sin R1 / tan DD) = C5m ... the Main Side Backing Angle


Proof ... cos P2 = sin R1 sin SS

Proof of cos P2 = sin R1  sin SS
Joe Bartok