Timber Framers Guild

Hawkindale Angles

Posted By: Dave Gee

Hawkindale Angles - 05/01/03 12:24 AM

Has anyone attempted to validate the Hawkindale Angle formulas?

I have attempted to use them for obtaining irregular hip and valley angles and have found that they didn't work for me.In my first attempt I used 7/12 (30.25°) and 12/12 (45°). DD+D = 60.50° not 90°. SS and S were correct. I then used 8/12 (33.69° and 10/12 (39.8°), DD+D=90° however in this case SS was 61.92° and S was 23.11°.

The formula R1, hip or valley pitch angle, is Arctan(TanS x Sin D). That is incorrect it should be Arctan(TanS x CosineD),that formula is shown as the formula A7 for the plumb backing angle for a hip or valley.

I would be interested to know if anyone else has experienced difficulties with these formulas and how and if they have reformulated them.
Posted By: Ed Levin

Re: Hawkindale Angles - 05/01/03 09:02 PM

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.
Posted By: Dave Gee

Re: Hawkindale Angles - 05/02/03 11:43 AM

Thank you for a speedy reply to my questions concerning Hawkindale Angles. From your response it appears that my questions were not explicit enough.

Firstly I appologise for making a math mistake, not a formulation mistake in my question. To try and understand the formulas as detailed on this web site I used two examples, both were for irregular pitches, regular plans not an irregular pitch and plan.

After reading your reply it appears my confussion lies around the basic formulas.

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

My first example was as follows:

SS Main Roof 12/12 or 45°
S Adjacent Roof 7/12 or 30.2564°

DD = Arctan[Tan(30.2564)/Tan(45)]= 30.2564°

D = Arctan[Tan(45)/Tan(30.2564)] = 59.74358°

SS= Arctan[Tan(59.74358)/Tan(30.2564)]= 71.20782°

S = Arctan[Tan(30.2564)/Tan(71.20782)]=11.22672°

R1 = Arctan[Tan(S)xSine(D)
= Arctan[Tan11.22672xSine59.74358]
= 62.54703°

Example 2:

SS Main Roof 10/12 or 39.80557°
S Adjacent Roof 8/12 or 33.69007°

DD= Arctan[Tan33.69007/Tan39.80557]= 38.65983°

D = Arctan[Tan39.80557/Tan33.69007]=51.34007°

SS= Arctan[Tan51.34007/Tan33.69007]=61.92739°

S = Arctan[Tan38.65983/Tan61.92739]=23.10646°

R1 = Arctan[Tan23.10646xSine51.34007]=18.42659°

Based on the above formulas and calculations the hip pitch angles for both roofs differ from those calculated from the more traditional methods.

A 12/12 - 7/12 roof would have a hip pitch of 26.74°

A 10/12 - 8/12 roof would have a hip pitch of 27.50045°

However if the pitch angle 'S' in the formula R1 was changed to the actual pitch angle of the adjacent roof them the R1 formula would produce the correct answer.

This raises the question what do 'SS' and 'S' really mean. If in the previous calculations the known roof pitches were used instead of the calculated pitches then different angles are created.

In a further attempt to try and understand the Hawkindale formulas I notice that some of the formulas do not account for the adjacent angles in hip and valleys of irregular roof and other formulas seem to have been omitted, for example the edge cut angle at the top of a hip or valley rafter and the top edge cut of a jack rafter or is the formula P2 to be used to obtain them.Looking at the drawing of the intersecting roof am I right in assuming that the formula P2 could also be Arctan[Cos SS/TanDD]
Posted By: Ed Levin

Re: Hawkindale Angles - 05/03/03 10:42 AM


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.
Posted By: SBE Builders

Re: Hawkindale Angles - 04/26/13 08:51 PM

Originally Posted By: Ed Levin

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.

This is an old topic/post. However, anyone looking for plan angles formulas for deck angles greater than 90 might want to take a look at this page for deck/eave angles greater than 90 with unequal pitched roofs.

Ed's trigonometric formulas only work when the unequal pitched roof have an deck angle somewhere less about 120. The cosine of an angle greater than 90 is always negative. Also, the plan angle formula should check to see if both calculated plan angles add up to the given deck angle.

Bevel angles for three dimensional connections

Martindale - plan angles

Frank L. Martindale had the correct plan angle formula along. It's similar to what I use.

One of the problems with roof framing trigonometry, is unconstructable roof framing.
unconstructable roof framing

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