Dave,

You raise a good point with your main/adjacent question. Every roof slope has its own entire set of Hawkindale values. For so-called main roofs, we call the pitch SS, the deck angle DD, and we give the suffix ‘m’ to most other variables. For adjacent roofs, pitch is S, deck angle DD, and suffix is ‘a’. Thus for the main roof

P2m = Arctan [Cos (SS)/Tan(DD)]

while in the adjacent roof

P2a = Arctan [Cos (S)/Tan(D)]

A guideline of the Hawkindale system is that the terms of the equations for angles in a main roof are written using only main roof values, those for an adjacent roof use all adjacent roof values. The only angle that does not get a main or adjacent designator is R1, the hip or valley pitch, which is common to both roof slopes.

Several vital variables govern any building project, three essentials being size, cost and quality of work. You can specify up to two of these, but not all three. For instance, building a large house on a limited budget will entail some sacrifice in quality. Alternately, if you want to do the best possible job without breaking the bank, you’ll need to shrink the space.

Likewise, in the world of compound geometry, you can set some but not all parameters. Typically the design process starts with the building plan, from which one knows the wall angle(s), and the roof pitches, or at least the main pitch, are also chosen a priori. Once you select the wall angle and common pitches, then the values of the deck angles and hip/valley pitch are not optional, but are predetermined by geometry.

So, to invoke my previous read on your first example, if the givens are the wall angle (W = 60.5) and the main and adjacent roof pitches (SS = 7:12, S = 12:12), then Euclid mandates that DD = 38.9751°, D = 21.5249°, and R1 = 20.1484°.

Please note that the simple formulas for determining deck angles from roof pitches

DD = Arctan[Tan(S)/Tan(SS)]
D = Arctan[Tan(SS)/Tan(S)]

do not work for irregular plan, irregular pitch roofs and you need to use the full-blown equations:

DD = Arctan[Sin(W)/( Tan(SS)/Tan(S) + Cos(W) )]
D = Arctan[Sin(W)/( Tan(S)/Tan(SS) + Cos(W) )]

For a given roof, main and adjacent pitches have unique values. Once you have chosen SS and S they are fixed, and cannot be back-calculated by formula.

These principles also apply to your second example, as they do to all compound roofs.

You are correct that your 10:12 – 8:12 regular plan roof would have a hip pitch of 27.50045°. But the hip pitch of a 12:12 - 712 roof would only be 26.74° if it too were regular in plan (W = 90°). Given a stipulated wall angle of 60.5°, the resultant hip pitch is R1 = 20.1484°.

To reiterate, you cannot randomly choose a path through the Hawkindale forest. Given starting points follow predictable routes to predetermined ends, the most typical being the following, in order of frequency:

1. Single roof pitch hip or valley roof over walls at right angles: SS = S, DD = D = 45°, W = 90°. These are givens, remaining angles are determined by calculation.

2. Regular plan, irregular pitch roof. W = 90°, SS is not equal to S, DD & D determined by formula in terms of W, S & SS

3. Regular pitch, irregular but symmetrical plan. Found in polygonal roofs where SS = S, W is not equal to 90°, DD = D = W/2. Usually either common (SS) or Hip (R1) pitch is given.

4. Irregular plan and pitch. Infrequent, characteristically found in a hip or valley roof on an odd corner lot where wall intersections are non-orthogonal. Presumably roof pitches are chosen, deck angles are then calculated in terms of W, SS and S.

In my experience, the Hawkindale angles are sufficient for layout and joinery of most of the roofs that most of us encounter most of the time. When the weirdness factor goes off the scale (out of plumb walls and non-planar roofs, out-of-level and/or crooked plates and ridges), then you have left the reservation and are on your own.