John K Jordan
>>>...one half of 180 minus 72, or 54 degrees. The angle setting on a miter saw may be the complement of this angle, 90-54, or 36 degrees.
It might be bit easier if you consider that 1/2 the angle of the "pie slice" would be 36 deg, the angle to cut on the end of the board. But of course, there are multiple ways to get the same answer. (Which is great since that's a good way to check the logic and the arithmetic!)
As for the numbers:
Calculating the board length and width based on the width of the wheel rim would take a couple of steps. One way I can think of to get the total length of the of the board (the long side of the segment) is to calculate the chord tangential to the outer radius of the wheel using the pythagorean theorum, then calculate the width of the board with trig using the cosine of the cut angle and the rim width as the hypotenuse of a right triangle.
Instead of the math, one could use a big compass or trammel points to draw the wheel actual size (or maybe 1/2 size), step off the five segments, then measure the drawing to see what size board to use.
Or an extremely easy way: use a segment calculator. Woodturners who make segmented pieces figure out this for the ring on every layer. I bought software to calculate the segments for every layer of a shape I can draw, but there are free on-line calculators such as this one:
BTW, if you use Win 10 there should be a scientific calculator hiding in the windows calculator under the menu in the upper left corner.