Dave Gee tested the Hawkindale Angle formulas on a couple of sample roofs. I’m not sure where he went wrong, but thought it worthwhile to work through the math for the two compound roofs under consideration.
Gee’s second example is the simpler of the two. It is a regular plan, irregular pitch roof. That is the eaves meet at right angles, but the main and adjacent roof pitches are different.
A couple of definitions before proceeding: SS is the main roof pitch, S is the adjacent roof pitch. The Wall Angle (W) is the angle in plan between eaves or ridges. It is the sum of the two Deck Angles (DD and D), where DD is the angle between valley & main ridge or hip & main eave, and D the angle between valley & adjacent ridge or hip & adjacent eave.
Mr. Gee specs 8:12 and 10:12 pitches for his second example. I’m not sure what he means when he says “however in this case SS was 61.92° and S was 23.11°” since by definition SS = 8:12 and S = 10:12 or vice versa.
Since it makes no difference which roof is main and which adjacent, let’s go with the following assumptions
Wall Angle W = DD + D = 90°
Main Pitch SS = 8:12 = 33.6901°
Adjacent Pitch S = 10:12 = 39.8056°
For regular plan roofs, Pythagoras tells us that the run of the hip/valley is the square root of the sum of the squares of the two common runs. If we take the rise as 10, then the run of the 8:12 roof is 15, that of the 10:12 roof is 12, and the hip run equals
Sqrt(15² + 12²) = 19.2094
By definition, the sine of the deck angles (DD or D) is the ratio of common run to hip run. Therefore
DD = Arcsin(15/19.2094) = 51.3402°
D = Arcsin(12/19.2094) = 38.6598°
Regular plan deck angles can also be found using the following standard Hawkindale formulas:
DD = Atan[Tan(S)/Tan(SS)] = 51.3402°
D = Atan[Tan(SS)/Tan(S)] = 38.6598°
Finally, rise over run gives us the hip or valley slope
R1 = Arctan(10/19.2094) = 27.5005°
Or, alternately, the Hawkindale formula for calculating the hip or valley slope is
R1 = Arctan[Tan(S)/Sin(D)] = Arctan(Tan(SS)/Sin(DD)] = 27.5005°
These results can also be confirmed via developed drawing (unfortunately illustration is not possible in this format).
Mr. Gee’s proposed formula
R1 = Arctan[Tan(S) x Cos(D)]
yields Hawkindale A7a = 33.0350°, which is not equal to the hip/valley slope.
For his first example, Dave Gee chose an irregular plan, irregular pitch roof where
Wall Angle W = 60.5°
Main Pitch SS = 7:12 = 30.2564°
Adjacent Pitch S = 12:12 = 45°
Here the formulas for deriving deck angles from known pitches and wall angle are a bit more complicated:
DD = Arctan[Sin(W)/( Tan(SS)/Tan(S) + Cos(W) )] = 38.9751°
D = Arctan[Sin(W)/( Tan(S)/Tan(SS) + Cos(W) )] = 21.5249°
By the way, note that when W = 90° these last two formulas reduce themselves to the standard regular plan deck angle equations given above.
Plugging the relevant values into the Hawkindale R1 formula, we find a hip/valley slope of
R1 = Arctan[Tan(S)/Sin(D)] = Arctan(Tan(SS)/Sin(DD)] = 20.1484°
All of these results can also be arrived at geometrically. Indeed geometric drawing is the foundation on which all mathematical treatments of compound roofs are built. Look for an article on this subject in an upcoming issue of Timber Framing.