Enter the Octagon

Is it time to talk about cheesy films from 1980? Am I a Chuck Norris fan? No, and no. I do remember seeing that film when I was in the 10th grade, but somehow I have absolutely no urge to watch it again. Now, old Shaw Bros. Classic Kung Fu movies, classics like The 36th Chamber of Shaolin, well, that’s an entirely different story – I’m addicted!

The octagons I want to talk about today are known across the US. Octagons, in fact, were very much in vogue in the middle of the 19th century. In 1848, through the publication of his book The Octagon House: A Home For All, or A New, Cheap, Convenient, and Superior Mode of Building, Orson Fowler created a craze in the for Octagonal houses (<– link).

Here’s a mugshot of Orson:

 I just got Fowler’s book through the local library system (those communists!) so it’s a bit premature to put up a review of the man’s work quite yet. He was also a believer in phrenology, that debunked ‘science’ primarily focused on measurements of the human skull, and was the publisher of a magazine entitled the American Phrenological Journal. Just because some bright people have clung to entirely erroneous beliefs however does not mean we should automatically discount everything for which they advocated. Like octagonal houses for example, which may have some merits. The town of Fowler, Colorado was apparently named after the gentleman, so he can’t have been all bad (?).

Some of the houses built upon Fowler’s designs definitely have their points of interest:

As I just acquired a copy of Fowler’s book, and have not had a chance to tour any octagonal houses yet, it is a bit premature to offer a review of that area of architecture at this juncture.

I do want to talk about octagons today though – specifically the type that are seen as bump-outs on many house walls. In New England, octagonal bay windows are virtually de-rigueur:

The above photos I have snapped on the drive home from my shop. One doesn’t have to search especially hard to find them around here.

Notice how varied the roof shapes are – looking just at the hip rafters, we have some with configurations where the rafters meet one another on the wall:

Based on casual inspection, I would say about 10~15% of the octagonal bump-outs I see are of that type.

Far more common are the type where the middle roof plane’s hips go straight back into the wall – here’s an example shamelessly grabbed from the web, though there are examples of the same in the preceding set of pics:

I’d estimate that about 60~70% of the octagonal bump-out roof forms in this area are of the above type. Too bad.

And finally one sees, about 10~15% of the time, versions where the hips come together to some degree as they ascend, without fully converging:

Given that the bump-outs invariably incorporate three wall planes of the octagon only, the converging hips are the correct way to do things, however there is much inconsistency in the way they are done. sometimes one gets the distinct impression that the carpenters doing the work weren’t quite sure how to bring the hips of the octagonal roof together on the wall, so they fudged things a bit. Oh, the client will never notice. You know, driving by on the road, that’ll look just fine. Let’s get on with the next thing.

Now, I stated rather brashly above that I believe there is a correct way to do things, and I’d like to delve a little more thoroughly into that in this post. I think what constitutes ‘correct’ in cases like this is not associated to artistic whim, but rather to a consideration of geometrical particulars. When you think about it, an octagonal bump-out is akin to having an octagonal cylinder, with an octagonal prism for a roof, occlude with the wall of the building:

The cylinder with roof is in white, while I have represented the wall with a grey vertical plane.

More common, it would appear, is that the octagonal cylinder in question is actually stretched on one axis so as to allow the middle plane to be wider, thus allowing for a wider window:

You can see in either case that the roof of the octagon would present hip rafters that were converging:

Now, most fortunately, figuring out how to lay out the hip rafters on the wall plane is not especially challenging. It can be done without any mathematics using full-scale drawing on the floor. All we need to do is draw the octagon and then mark a line across it indicating the plane of the outside surface of the wall – as I have done in the following sketch:

 One could size an octagon, I might add, based on the width of a given facet, or based on the run of the octagon hip, or, as I have done above, by the projection from the wall at 26″.

With the line in place across the octagon’s plan, all one would need to do is measure the distance between the hips at the wall line and one would have the required distance to mark on the plane of the wall.

Let’s say, however, I didn’t have free floor space to lay out upon, or perhaps I wanted to do the work on the back of a napkin using a simple calculator function. We know that an octagon is 8 sided. 360˚ divided by gives 45˚, and the hips would bisect that angle, giving us 22.5˚. The octagon in plan therefore has a relation between the run of a hip and the run of the common of 22.5˚. With a given 26″ projection of the octagon from the wall plane, we can use tangent on the calculator, the relation between opposite and adjacent:

22.5˚ (TAN) = 0.4142135….

That’s a convenient value to remember actually, since the octagon features 45˚ turns, and the relation of a 45˚ unit triangle is 1:1:√2. Since √2 equals 1.4142135…, all we need to do is subtract 1 from that and we have the tangent of the 22.5˚ at 0.4142135…..

Our adjacent in this case is 26″ long, so we multiply the tangent value by 26 times (since the calculator works upon the basis of a unit run of 1, not 26):

(0.4142135….) x (26) = 10.769553….

Here’s a depiction of what we have found in determining the length of the measure opposite the 22.5˚ angle:

In this case, the facet of the octagon measured 36.769, so we subtract the 10.769553 value we just found from that twice (that is, from both ends of the facet) to obtain the distance between the centerlines of our octagon hips where they meet the wall, namely 15.230447″.

With that measurement established, all we need to do is establish the height on the wall above the octagonal bump-out’s wall plate where the roof plane meets the wall. In this case, I have made, as is common in New England, the roof at a 45˚ angle, or 12/12. That means that the distance the octagon bumps out horizontally from the wall is going to be the same value as the roof would climb to at the place where it meets the wall, 26″:

We therefore in this case measure up 26″ from the wall plate:

Once we have drawn the line on the wall above the octagon’s plate, we can then establish marks for the width between the hip centers on the wall as we had determined earlier, that 15.203..” measure:

And that is that connect the dots and we would know where to cut the sheathing on the wall to allow space for the octagonal roof surfaces to intersect:

So, there we have it, though I must confess I’ve only scratched the surface when it comes to polygonal geometry as it relates to architectural forms, and more particularly when it comes to joinery work involving polygons. One day I’ll be digging into that in more detail, likely with an essay for the TAJCD series. And for sure, the online study group will be delving into such material in the future as well. For now, I must remain an observer, wishing most of the time as I travel about that more carpenters would take an extra minute to figure this sort of stuff out before letting loose with the saw and nail gun.

All for today – thanks for coming by!

2 Replies to “Enter the Octagon”

  1. Chris

    There are lots of octagonal bays here in Cleveland, most from the 20's,and even then there was some confused roof framing. One roof, on a nice stone house, has the roof planes curved to meet the hips! The eaves are straight. Audels says the run of an octagonal hip is 13, but its probably safer to bisect the angle and measure up the rise, as you suggest. Funny how quickly things get complicated…


  2. Hi Tom,

    thanks for your comment. I'd be interested to see what that roof you mentioned looks like – it's a bit hard to picture.

    As for the run of an octagon hip being 13: if you followed the math I did above on the plan view of an octagon, note that for a run of 1 and given a 22.5˚ angle in plan, a little button pushing returned an 'opposite' measure of 0.41421356…

    We have two legs of the 22.5˚ plan triangle then, the adjacent at 1, and the opposite at 0.414213… If we used Pythagorus to work out the hypotenuse length for that plan triangle, we would find it equals 1.0823922…

    That hypotenuse is the run of the octagon hip. Since in Western framing one normally takes 12″ as the common rafter run on the framing square, if we multiply 12 by 1.0823922.. we obtain 12.988706.., a value is less than 1/64″ from 13″. So in practice it is said that the run of an octagon hip is 13.

    You're right though that things can get complicated. In the case of the problem of layout out of the octagon roof profile on the building wall, illustrated in the post above, there's really no need for any complexity.


Anything to add?

error: Content is protected !!
%d bloggers like this: