What are you making
A picture frame or a box is four sides. A hexagon planter is six. A round-looking ring is just a lot of short sides.
Measure the angle inside the corner where the two pieces meet. A square corner is 90. A gentle turn is more than 90.
Enter how many sides the frame has.
Cut a test corner in scrap first and hold the two pieces together. If the joint gapes on the inside, the angle is a hair too steep; if it gapes on the outside, a hair too shallow. A degree or two off shows up fast across four corners.
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A frame is only as square as its worst corner, and a miter that is a degree off turns into a visible gap by the time you have cut four of them. Tell this how many sides your frame has, or the angle of the corner you are turning, and it gives you the angle to set on the saw.
Frames and rings
For any flat frame with equal sides, the miter on each end is 180 divided by the number of sides. Four sides gives 45 degrees, the picture-frame angle everyone knows. Six sides gives 30, eight gives 22.5. Both pieces that meet at a corner are cut at the same angle, mirror images of each other, and together they close the corner. A ring that looks round is really just a frame with a lot of short sides, and the same rule sets it.
Any corner
When the corner is not part of a regular shape, measure the angle inside it and switch to the corner mode. The miter on each piece is 180 minus that angle, split in two. A 90 degree corner comes out at 45, a 135 degree corner at 22.5. This is how you wrap trim around a bay window or a corner that is not square.
Reading your saw
Most miter saws measure the cut as how far the blade swings away from square, so a picture frame reads 45 on the scale, which is the main number here. A few tools, and some protractors, measure from the blade sitting straight across the work instead, and for those you want 90 minus the miter, shown next to it. When in doubt, cut a corner in scrap and check it before you touch your good stock.
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