Making splayed miter jointsA friend, Joel, recently asked me about how to calculate the miters for making a sort of tapered cylinder out of wood segments. For example, for a straight eight sided cylinder, the pieces need to be bevelled at 22.5 degrees. But when the cylinder is tapered, that same 22.5 degree bevel leaves gaps on the inside.
Joel works in a high school wood shop, so first he turned to the math teachers at his school, but they weren't able to give him an answer. When he showed me the high school wood shop he works in, he asked me if I knew anything about how to work it out.
I knew about the problem, but hadn't had a reason to work out the math for it so far. Looking around the web, I couldn't readily find a site that already had it worked out, so I thought it might be a fun thing to cover on this site.
I worked out the math for it. It's a lot of trigonometry. All based on high school math but, at the same time, no doubt too much for most high school math teachers.
A number of people have asked about the math for the claculations. I worked it out on paper, and then made a Sketchup drawing to verify my calculations. I also typed the formulas into the Sketchup model so I wouldn't lose them. You can enlarge the image at left to for a better view.
You can also download the sketchup CAD model for this drawing to have a closer look (you need to install the free Google Sketchup to open the CAD model)
I had cut the test pieces using my miter saw. But I'm really no fan of using a miter saw, especially for small pieces, where it's difficult to secure the workpiece against the low fence. For my second piece, I went back to using the table saw.
Note that the miter and blade tilt angles apply the same way on the miter saw as on the table saw.
Setting the miter angle. Experience has shown that the scale on the miter gauge that came with my Delta hybrid saw is surprisingly trustworthy. I'm always amazed how people swap these out for the costly Incra ones that are only made of bent sheet metal.
But if you want to second guess and check, a good way to get accurate angles is to use one of those rafter gauge type squares to set a bevel gauge. The rafter square is one piece, with a large angle scale, so there's little to go wrong. Just push the bevel gauge against the inside corner and set it to the scale on the long edge.
I wanted this 10-sided shape to have a diameter of 30 cm (flat to flat), and using my table, I had worked out that the longest edge on the pieces needs to be 9.7 cm long. Here's measuring the first piece.
Now flipping it over, transferring my mark to the other side, and lining it up with the kerf in my sacrificial fence. I clamped a stop to the fence so I could just push subsequent workpieces up against it.
Cutting the pieces. I used a quick grip clamp to clamp the workpiece against the fence. Although, in retrospect, that caused the workpiece to rock up slightly, which may have contributed to some inaccuracy.
Ideally, I'd like to apply a clamping force that is centered on the area of the joint.
Many people prefer to clamp joints like this by applying tape to the outside, but it's impossible to get good clamping force with tape. I want to make sure these are good and solid joints.
What bevel angle to cut on the scrap piece? Exactly the same angle as the joint, so I just ripped some scrap wood with the angle that I already had set from cutting the workpieces.
This photo makes it look too easy. It's actually quite fiddly getting these together. One of those one-handed quick-grip clamps makes it much easier, but, unfortunately, these quick grip clamps can't apply very much force, so I don't consider them suitable for glueups.
A piece of tape applied to the outside of the joint would probably have made it easier to keep it aligned, but I'm not used to working with tape in the workshop, so I didn't think of it at the time.
The joint is a simple butt joint. When I did my glue experiments, I found that a well glued butt joint can be surprisingly strong. Butt joints need a lot of glue though, because the end-grain really soaks it in.
Four pairs of two glued up. I was about to glue the fifth pair of two when I realized -- oops, that would not be a good idea. I want to glue the pieces to two halves, then joint the halves, so I need to end up with two parts of five segments each.
With ten miter cuts on each half, any angle error in the miter joints will add up ten times at the gap! So I wasn't expecting to get this perfect, although I had hoped for better. My workpieces tended to lift up off the table saw when I clamped them to the fence. That may have added some error.
By clamping the joints one joint at a time, all the error accumulated to just two joints. Had I clamped it together using rubber bands or tape, the error would have resulted in small gaps in random places instead.
Same method again with the beveled pieces of scrap to clamp to. But getting it to align is a tricky thing to do. You may want to curse a bit when you do this. Not that cursing helps any, but maybe it makes you feel better :)
I ended up turning it into a bowl.
Testing glue types
A butt joint can be surprisingly strong.
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