Table Saw Bevel Maths?

Question

I'm looking to make Triangular & Hexagonal shelves, but I'm getting confused by some of the maths involved for the bevels. To make a hexagon I need a 120 degree angle, but I need 2 pieces to make that angle, so I do: 120/2 = 60 degrees each.

So I would think that I can set my bevel gauge to 60 degrees on my table saw (Hypothetically, I'm aware that my bevel doesn't go that far and that I can use a jig to hold the workpeice vertically to do the complement of 30 degrees, effectively making it 60 degrees)

I'm having similar issues for the equilateral triangle shelves. I've seen people cut their straight stock at a 30 degree bevel and wind up with angles that don't look right, such as:

Here, you see that the shelf is not assembled at the edges, and is instead on the face. This doesn't look right to me, and seems to imply that the person didn't know how to cut the angles properly. I've seen proper bevels cut and they look really nice, even at non 90 degree bevels, such as this one from Foureyes:

How can I resolve these issues? What bevel should I cut my straight stock on, so that I can make these shapes?

EDIT 0: In general, if I want to cut a 30 degree angle in Cartesian space, do I set my bevel gauge to 30 degrees?

EDIT 1: Stranger yet, I've seen the hexagons be cut at a 30 degree bevel by this video (https://youtu.be/_sQzIbgQjig?t=38), and my intuition tels me that a transformation function is used to convert between Cartesian space and table saw bevel space. He needed a 60 degree angle in Cartesian space so he took the compliment of 60 degrees and obtained an angle of 30 degrees, and this is what he set his table saw to. Is this thinking correct?

Answer 1

Does it help to think about it backwards (and tongue in cheek)?

Use your equilateral triangular shelf as a starting point. Pretend Ms. Plato, a retired carpenter down the street has gifted you with the perfect wooden equilateral triangle of boards like your photo (rather than lines with no width). Unlike the picture, Ms Plato's joints are so precise and perfectly finished that you can't detect where the boards came together. She may have used a piano-black, high gloss finish (How do I achieve a "piano black" high gloss finish on wood?) or not.

Those boards in the finished triangle actually form two triangles, with a larger triangle formed by the junctions of the outer faces of the boards used, and a smaller "concentric" inner triangle where the inner faces meet.

Now, you've been allowed to choose a medium quality crosscut saw: Japanese or "European" being your choice. Here's where it gets dirty. You're going to saw apart that ideal triangle. The only rules are a) that you need to use a Moxon vise constructed from a YouTube video and b) you have to start sawing in a straight line at each vertex on the inner triangle. For example, when you start sawing at inner vertex B (with your labeling preference) you can point that Dozuki in any direction you like to make the cut. (You're not allowed to have a "hook" after the cut :)

Notice that after making a cut from each vertex and sweeping up the sawdust that you have three pieces of wood. The angles on the end of those boards are probably all different. Maybe Steve Ramsey would call it "funkily" different. The only thing that appears to be true upon measuring is that the angles of adjoining faces add up to 60 degrees, and that happens for each of the three junctions you have.

You and I would probably like the 30-degreeers best, but not everyone would share our aesthetic. If people think the way we do and were building that triangle, they'd probably setup a table saw to cut those 30 degree angles, using all the real-life techniques that woodworkers have in their apron to creep up on the cut, do it accurately and consistently, adjust to minor mistakes etc. But some may prefer the triangle of your photo.

I agree with the "try it" gang (see how close you can get to the desired angle, use the 30 + 60 = 90 approach, do a good job dimensioning the boards you'll use, etc.) There are various tools you could use to determine if the angle you cut (afterwards) is what you wanted.

I also suspect for polygons of increasing numbers of faces, mistakes in the degree of the cut for each face would compound more, and maybe a person would say do all but the last face, then evaluate what you had to see what was truly needed for the last face, rather than just cut the ideal.