There are 360 degrees in the full range of direction from the center of an object, but looking from the perspective of each piece used to form that object is different. The minimal number of degrees of angles to form a shape is 180 degrees... A triangle. Add 180 degrees and you can make a square. Add another 180 degrees and you can make a pentagram, Add another and you can make six sides, etc.If any readers have a simpler or more direct means to do this calculation, or if you have a website to suggest offering these kinds of carpenter's math solutions please share.
To make a 10 sided shape, you know there are seven sides more than a triangle, so you add 7 x 180 degrees (1260 degrees) to the base of 180 for a total of 1440 degrees. Divide that by 10, then by 2 and you know that each piece should be cut on each edge at a 72 degree angle. This should work for any cylindrical assembly, but remember that your table saw reads 0 degrees when the blade is at 90, so if you are using the gauge on the saw you have to do some subtraction from 90 degrees to set the arbor at the correct angle.
A brain is a terrible thing to waste. My thanks to Alex for putting mine to use.