Miter Angles for Two Different Board Widths - methods in a.b.p.w


One of my sons is making a coffee table 18" tall, 32" wide and 48" long. The catch on his idea is that he wants it to be 4 inches thick - in black walnut! When I COULD get Claro Walnut in the size and thickness "we" now need, HE had a completely different design in mind - and in a light wood. I'll skip the "Oh, and I want an inset box in the top" for another time.
So - I had a couple of sheets of 3/4" black walnut ply - both faces with "edge joined, bookmatched boards" look. (This stuff is going for $182 a sheet around here) To get the the thickness, I suggested making three torsion boxes with the black walnut ply for the skins, poplar between them.
Making mitered corners torsion boxes is a bit beyond his furntiure making skills level - and probably mine as well. We'd do a normal torsion box for the top and leave the outside face of the sides long (see ASCII diagram belOW)
+-------------------- +-+--+---------------- | | : | |---+--------------- | |-------+-+-------- | | ........| | | | | |
3/4 x 3/4 black walnut to cover the ends of the ply that showed and a black walnut face frame to hide the sides and it'd look pretty much like it was solid walnut - almost. If we butt joined the face frame parts, end grain would be showing somewhere. THAT was not acceptable to either of us.
Now had we made the sides and top torsion boxes ALL four inches thick life would've been much easier. Alas, since the underside of the top would only be seen if you layed on the floor and looked up - and to save ME from having to use part of another sheet of the $182 a sheet, Good Walnut Ply, we went with a piece of quarter inch baltic birch ply for the bottom of the top torsion box. That missing 1/2" would come back to bite us in the ass later - which is what my posts to alt.binaries.pictures.woodworking is about.
You see, when you go to miter the corners of two boards which are NOT the same width, the miter angle is NOT 45 degrees on both boards. Hell, it isn't 45 degrees on either board.
Now, were it my decision, I'd go with 4" wide boards for the face frame, cut them at 45 degrees and be done with it. No one's going to be the wiser so what the hell.
"But then you'd feel a lip where the bottom of the face frame and the bottom of the top torsion box meet. And besides, I've already ripped the top face frame parts to 3 1/2" and the sides parts to 4". I don't want a lip and I don't want to go get more walnut boards. Let's JUST figure out what the miter angle (sic) is for what we've got."
THAT is the subject of the posts to a.b.p.w.
charlie b (the guy with a son who has expensive taste and very specific ideas for what his "piece" should look like)
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SNIP...

The angles you're looking for are not hard to calculate. Use a little simple trig and you're done.
For most miters, both sides have the same width and therefore the tangent is 1 and the arc tangent is 45 degrees. For your project where one width is 3/4" and the other is 1/4" there are two tangents you need to calculate
3/4 divided by 1/4 = 3. The arc tangent of 3 is 71.565 degrees. For the other angle you have 1/4 divided by 3/4 = 1/3. The arc tangent of 1/3 is 18.435 degrees.
Now I suspect that your plywood isn't really 3/4" and 1/4". It's most likely something close like 23/32" maybe 7/32 or there abouts.
You could measure your wood and do the math. Or you could make yourself a triangle using the same ratios as your wood and then use that triangle to adjust the angle on your saw to cut the actual miter.
For example, assuming 3/4" and 1/4" as your thicknesses.
Cut out a square using scrap wood. Let's say about 12 inches. Now measure 9 inches from the corner and make a mark on the edge. This represents the 3/4" wood. Measure 3 inches from the same corner and mark the other edge. This represents the 1/4" wood. Now draw a line connecting the two points and cut along that line. You will now have a right triangle with the legs being 3 inches and 9 inches. The two angles on this triangle are the two angles you want to cut your wood at in order to get the miter you want.
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John Cochran wrote:

snip
That, along with a little Pytharogus, is what my post to a.b.p.w. was about - with three "pages" of illustrations and solutions with numeric examples.
The idea of scaling things up to reduce marking and measuring erros is a good one though. The larger scale parts would make setting up the miter saw more accurate. Thanks.
charlie b
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