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When you’re learning something new, it’s hard to sift what you need to know out of what you assume you need to know. You can spend a lot of time and money on something that doesn’t move you any closer to where you want to be. There are plenty of these traps on the way to becoming an accomplished woodworker, and it doesn’t help the beginner that there are plenty of pundits and purveyors of tools who will gladly lead you down a wayward path. Angles other than 90 or 45 degrees are a good example of this.
The practice of measuring, laying out and cutting angles in terms of degrees is rather new in woodworking. Old woodworking texts discuss angles in terms of rise and run; make a mark a certain distance horizontally and another mark a set distance vertically. Connect the marks with a line, cut and trim to the line and your done. What’s important is that two pieces of wood fit nicely together when assembled. 
The pile of tools in the photo represents a progression of devices used to measure angles. Most of them claim an advantage due to the ability to resolve angle measurements in finer and finer increments. But the one tool that actually gets used is the simple bevel with no numbers at all. The reason for that is it works reliably and accurately, quickly and without confusion. Just because someone can make a device that displays angles in hundredths of a degree, doesn’t mean there is an advantage in using it.
So here’s one of the secrets of what we do when we publish plans in Popular Woodworking Magazine. We draw angles by making marks and connecting lines, when we design with our CAD program and when we work in the shop. If someone wanders by while we’re in the shop and asks us what the angle is in degrees, we give them a lengthy discourse on why they don’t need to know that. Then at the very end, we let our computer measure the angle and we throw the angle measurement into the drawing. Because we know that someone will ask.
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Numbers do have their place.
I want to do a coopered door. Due to my design needs it has to have 19 facets. What is the rise and run of my angles? Quick, I’m still waiting! <G>
Is it better to spend an hour doing the layout to find out or do I just divide the angle of the arc by 19 and be done with it? Then I can set my drafting triangle to the correct angle draw the angle and use it to set my bevel (or even faster do it on the computer and print it out).
You can waste a lot of time on either method if you don’t think through the problem first to figure out which is better for this situation.
The biggest problem is so many woodworkers are slaves to one side of the issue or the other. As with cutting tennons you can do it by hand or use power tools. Just remember there are advantages and disadvantages to either method. The smart woodworker is the one who knows which method is the best for the situation at hand.
I just remember a comment from Norm Abrams’ "Measure Twice, Cut Once." He states that whenever possible, it is best and more accurate to not even measure. That may entail holding a piece in place and marking to length in place or for angles, it may mean holding to pieces at an angle, letting them overlap – mark the intersection points on each side and then connect the dots. When you cut to that line, the pieces are virtually guaranteed to fit properly.
I’m just wondering if we are putting so much trust and time in tools & technology that we just don’t spend the time with our hands and mind. There is just no substitute for quality time in the shop. A pencil, paper, rule, and a little common sense. My old grandfather with a 3rd grade education taught me more over 50 years ago than 90% of all the ads for new tech that are so called precise instruments. Time in the shop is the best teacher especially if you make a few mistakes, they seem to be my best teacher.
That said I have to get back in the shop enough technology for today.
Chris: if you cope the moulding, which arguably you ought to for most interior mouldings, then you don’t need to halve the angle. If you’ve got an outside corner that you need to have the moulding meet around and it’s not at 90° (when is it ever) you can use a sort of protractor that has a middle blade that automatically halves the angle, but that doesn’t come up all that often.
There are various ways to represent an angle. We commonly would describe a dovetailing angle as 1:8. In doing so, we are simply specifying the slope or tangent of the angle of interest, rather than giving its value in degrees. Which is more convenient depends on how it will be laid out, and/or what scales are on our machines. But it’s really the same thing.
Just curious about how you handle the case of making miters to fit an arbitrary angle. I suppose you could use a bevel gauge to transfer the angle to a piece of paper, bisect the angle geometrically, then use the bevel gauge to transfer the result to a power miter saw. However, I think it would probably be faster and more accurate to take a reading with a digital protractor, reset the protractor to half the angle, and then use that to set the saw.
Robert,
I agree 100%! When I was a younger man, I learned how to build houses by stick framing. We actually started with a pile of lumber! We would cut all the required wood (rafters, stairs, headers, etc.) on site and never used specific angles to assemble the home. "Field sense" applied. Every home was different, even though they were "trac" homes. Rise/Run was the guiding rule.
Even though I am many years removed from home construction, I still apply these principles to my woodworking. I couldn’t tell you what angle results from the layout, just that it looks good and it fits.
Chuck