### On the Slope

#### by Mathieu

In this post we will look at the differences between a roof

slope defined by a ratio or a roof slope defined by degrees. I have

often questioned why a carpenter would choose one method over the

other. Most carpenters who produce stick frames use angles in

degrees and this works well in combination with a miter saw. Timber

framers however, who use timbers of greater dimensions tend to use

proportions, although some layout methods rely on degrees as well.

I decided to compare both systems in the light of their usefulness

in practice and not only from theoretical point of view. Let’s just consider a

quarter of a circle, 90° in other words a right angle for our case

study. First we will look at a graphical comparison of both

systems. Below I have filled up two similar portions with different

increments. On the right side you see the increments

representing 10 degrees and on the left side we have increments

that represent proportions. In this example I have

used a run of 30 units since this is what I am used to when working

with a metric Japanese framing square or

*sashigane*. Don’t worry if you are not familiar

with this particular Japanese square since it could be any framing

square as long as it has the same scale on either arm. The only

thing that’s important to understand is that we are comparing

proportions to degrees. A roof slope would normally not be

described in a ratio to 30 as it is in this drawing, but instead to

10 (or 12 if you were using fractional inches). It is important to

note that if we use proportions/ratio’s the scale doesn’t matter.

You can use any system of measurement: imperial, metric, shaku,

Ell, or

come up with something yourself. A slope of 4/10 is going to be the

same no matter which unit of measure you prefer to use. This is

inherent to the mathematical essence of ratio’s. When we use

degrees a quarter circle is divided in 90° (a right angle). In this

case study we will limit ourselves to an accuracy of 1/2 degree

(for example 46,5°) , and to 0.1 of a *rise*

unit when we are referring to ratio’s (for example 3.5/10). Greater

accuracy than 1/2 degree is almost impossible to achieve since I

don’t know of any protractor suitable for woodworking that would

allow this. When using ratio’s we limit ourselves to the smallest

increment on our square. Below there is one example of a framing

square with a protractor. At first sight it seems well made,

stainless steel and etched markings. I have never used this one

myself but I know some carpenters who do and I don’t think this

particular model is built with any accuracy in mind. It is

impossible to use the short arm with a greater accuracy than 5mm

and this will not get the job done to my standards. I do not

understand the design of this particular square and therefore it

would not be my weapon of choice. The Starret protractor, as

shown below, is one I do use for carpentry and lay out purposes.

This is clearly a very accurate tool, originally designed

for machinists and I find it quite useful but I would not consider

it a mandatory tool for your tool chest. For the layout of large

timbers and roof carpentry it’s use is rather limited since it is

not designed for this purpose and it does not substitute a good

framing square. The framing square remains

*the* essential tool in carpentry layout but also in other woodcrafts as the

picture below illustrates. It depicts John Browne ,1660-1670, the instrument maker. Although he is not a carpenter the fact that he is holding a framing square and the platonic solids seen in the background illustrates that he was also a mathematician who studied Euclid. (Thanks to Sim Ayers for pointing out the correct information about this picture) The next picture shows the

typical tables etched in a modern western style framing square.

Note that this square has decimal inches instead of fractional

inches. Maybe it’s because I grew up in a part of the world where

measure is expressed decimally that I feel uncomfortable using

fractions. I used fractional inches for quite some years when

working in the US and never became a great fan of it. Just a

personal point of view, read *not of any
significance. *In the end a carpenter should be able to

adapt to any given situation and this includes a different measure

system. These are two sashigane

one with a metric scale and the other with a shaku/sun scale. You

can read more about them here.

The sashigane has two

sides one with a regular scale and the other with the square root

scale to lay out hip rafters for example. Let’s leave the numbered

pieces of steel behind and get back to our comparison of ratio’s

and degrees. We noticed that the degree increments are equally

spaced but the proportional increments are not. The linear distance

measured between the proportional lines on the circumference of our

circle becomes smaller. Here is another version of the drawing that

shows the proportions in relation to a square. There are actually many

more lines on the drawing, one for every mm on the short arm and

one for every 1/2 degree, but to keep the drawing clear they are

not shown here. Drawing this was fun, I couldn’t figure out an

efficient way to draw the lines of the proportional slopes and

ended up drawing many lines manually which was not only boring but

quite time consuming as well. I need to get better at this

since repetitive tasks should be very straight forward with CAD. If

you are interested in the process, which I am afraid you are not,

it goes like this. And no I don’t want to be annoying but I do feel

the compulsive need to share the procedure with you. Click

‘enter’ to repeat the previous command, hover the cursor close to

the previous line to select it’s ‘end point’ and start the new line

from there, type ‘1000’ for the length of the line, click

‘tab’ to select the dialog box for the angle of the line and click

at the reference point which was already established. This

procedure takes me about 4 seconds – that is if I do it correctly

and of course that’s not always the case and then I lost another

six seconds in between the lines. On top of that I could have

continued endlessly, in the true sense of the word, forever and

ever and ever. This brings us to the next observation. There is a

difference between both systems when we look at the amount of

divisions of our right angle. Employing an accuracy of 1/2

degree means we have 180 different angles to choose from when using

degrees as our unit of measure to define a roof slope. If we

employ proportions, in theory we should have an endless

amount of angles to choose from. The

*rise*in

our proportional method could have any possible length, even then

we are still able to define an angle smaller then 90° with it. The

angular value between 4 and 5 degrees is the same as the angular

value between 28 and 29 degrees, that is 1°. This is different when

we use the proportional method along the arms of a framing square.

The angular value between 8.3/10 and 8.6/10 is about 1° but the

angular value between 28.3/10 and 30/10 is also 1°! In one occasion 1° is

represented by 3 increments on the scale of the framing square and

in the other occasion 1° is represented by 17 increments. Very

important to keep in mind during the design process. When a

designer uses degrees for his roof slopes when drawing his plans,

he might have a tendency to come up with angles like 37,15°. But if

the carpenter uses ratio’s when laying out his roof, things can

become unnecessarily complicated. You could convert all the angles

to ratio’s but this is just another step where mistakes can creep

in and you end up with strange numbers which are unpleasant to work

with. I believe that a design should be made using the same

system that will be employed in it’s execution just for the sake of

simplicity. To convert a roof slope to an angle we divide the

numerator by the denominator which gives us the tangent value of

our roof slope. We then use the inverse tangent function (tan⁻¹ or

arctangent) to get the angle expressed in degrees.

Example:

3.5/10 = 0,35 tan⁻¹=

19,29°

or

37,15° tan = 0,756667783 x 10 =

7.6/10

In the next post we will continue to look at the

differences and reflect them to their practical value. The question

remains whether the method we choose will suit our needs on the

construction site?

∴

Great blog

I enjoyed reading very much thank you for posting

Regards

Andy Ryalls

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Great blog on roof framing geometry.

However, the drawing you refer to as a master carpenter is John Browne ,1660-1670, the instrument maker. He was not a carpenter. If you had the original image, you would see the 3 platonic solids in the drawing that would indicate he was a mathematician as well, who studied Elucid.

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Thank you for pointing out the correct information regarding the drawing. I will adapt the text accordingly.

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