The number of facets (sides) for your ring will determine how to perform the cuts necessary to make the ring.
For example, one can consider a square formed from 2x4s to be a ring. Each cut would be half of the 90° angle at the corners, or 45° set on the miter saw.
Jumping to double the above, a 22.5° cut on eight pieces will give you an octagon, 11.25° will give you a sixteen facet ring.
If threes are your thing, a triangle needs 30° cuts to make the 60° joining, while a hexagon with 120° angles will magically appear when your cuts are 60° on the saw.
Going another step to 12 sides means a cut at 75°, or 15° from square to the end.
The math is pretty simple.
360 divided by number of sides = interior angle. interior angle divided by 2 = cutting angle.
Be aware that it's important to track which side of the lumber is "outside" as flipping the wood incorrectly will result in a parallelogram instead of a trapezoid.
This would depend on what size of a ring, as well as how round you want it to be. Shorter pieces of wood, cut at smaller angles, would give you a more round ring, longer pieces of wood at a greater angle would give you a less round ring. Changing the angle would change the size of the ring as well as the roundness.