Precise Measuring of Angles for Segmented Bowls

I am a woodworker and recently, I have started to make Segmented Bowls. In order to do this, one makes a ring out of smallish wood segments that are cut to very precise angles. An error of even a tenth of a degree is multiplied by the number of segments - typically 8 to 12 or more and up to 144 for some very complex bowls. This causes the ring to not meet at the end. There are ways to compensate for this, but I prefer to get my angles more precise.

Initially, I purchased a high precision digital protractor. It was not high precision enough.
My first ring ended up with a gap of several degrees.  I viewed this experiment as a failure.

My First Sled

So I moved to geometry. That worked much better. Here is a photo with the clamps in place and a workpiece up against the stop ready to be cut.

Here is the same sled with the clamps removed and the distances marked:

So - what about the math? First of all, the base line (closest to the saw operator) must be perpendicular to the blade. That should be easy on a well-aligned saw.

Next measure the "Line parallel to base". I suggest metric system since the rulers are usually more precisely calibrated and also in decimal units.

To get "D", multiply the this distance by the Tangent of the Angle. In this example we are looking for a 12 segment bowl, so we can go to Google and enter "Tangent of 15 degrees" to get .26795.

Draw the lines on the wood and position the fence board. You should get an exact 15 degree cut.

Using Yardsticks

Last weekend, I attended the Segmented Woodturner's Symposium. There, I was introduced to Wedgies - a better way to make a segment cutting sled. This seems to be an improved technique. It uses two fences. The angle with the base line is not as important as the difference in angle between the two fences. I have not yet built one of these, but a major problem seems to be getting this precise angle. My solution again is using geometry - this time using yardsticks. 

Well, not exactly yard sticks - more like meter sticks. I bought three (that's right - three) 40" rulers from Harbor Freight. These are just over one meter long. I arranged them as follows using three clamps:

On the far end, I clamped the two yardsticks so that the 1 meter marks exactly lined up:

This time, I am looking for a thirty degree angle. So, I 

1. Cut the angle in half (30 / 2 = 15 degrees)
2. Get the Sine of this (Sin(15) = ,2588)
3. Multiply this by the aligned marks on the ruler (.2588 x 100 cm = 25.88 cm)
4. Double this (25.88 x 2 = 51.76 cm)

So, rather than trying to use the ends of both rulers, I opted for setting the left side at 10 mm and the right side for 61.76 mm.

I then measured this with my "High Precision" angle ruler

and it measured 29.8 degrees. Does this mean that my system was off by 0.2 degrees? NO! It means that my precision angle protractor is off by .2 degrees. Multiply this by 12 and we have a total error of 2.4 degrees. Not good enough for a closed segmented bowl ring.  (To be fair, the protractor is specified for .3 degrees, so it is within spec). How do I know for sure which is correct? 

Well, my wife's father was a printer and he left us a large metal 30 - 60 - 90 triangle which I think is probably pretty accurate. For years, my wife has insisted that we hold onto this as a souvenir of her father's business. Well, I tried that and it exactly fit in the angle I created using geometry.