Every flat frame follows one rule: the cuts have to add up to a full turn. Divide 360 by the number of sides to get how much each corner turns, then halve it because two pieces share the corner. That halving is the number you set on the saw.
Why it is 180 divided by the sides
A closed shape turns through 360 degrees no matter how many sides it has. A square turns 90 degrees at each of four corners. A hexagon turns 60 at each of six.
But each corner is made by two pieces, and each piece only contributes half of that turn. So the saw setting is half the corner turn, which works out to 180 divided by the number of sides.
Four sides gives 45 degrees. Six gives 30. Eight gives 22.5. Twelve gives 15. That is the whole calculation, and it holds for any number of sides down to a triangle at 60.
The interior angle is a different number
People mix these up constantly. The interior angle is the angle you would measure inside the finished frame where two pieces meet — 90 degrees in a square, 120 in a hexagon.
The miter is half the supplement of that, not half the interior angle itself. The calculator shows both so there is no ambiguity about which one goes on the saw.
If you are copying an angle off an existing piece with a bevel gauge, you are measuring the interior angle. Halve its supplement to get the saw setting.
Accumulated error is the real enemy
A quarter of a degree of error in one cut is invisible. The problem is that a four-sided frame has eight cuts, and if the saw is consistently off in the same direction, all eight errors add up in the same place.
That is why the last corner is always the one with the gap. It is not that the last corner was cut worse — it is where seven previous small errors have collected.
The fix is not more careful measuring. It is to check the saw against a test joint first, then cut all pieces without touching the setting.
The five-cut method
To find out whether your saw is actually square, there is a standard test. Cut a piece off one edge of a square offcut, rotate it, cut again, four times. On the fifth cut, take a thin strip off and measure its width at both ends.
Any error in the saw has been multiplied by four in that strip. Divide the difference by four and you have your true error, magnified enough to actually see.
This takes ten minutes and it is the highest-value ten minutes in any miter project.
Frames are not the only use
The same arithmetic covers segmented rings for round tabletops, octagonal posts, hexagonal planters, and the staves of a coopered column.
For a ring you also need the segment lengths, which depend on the diameter — the segmented ring calculator on this site handles that case.
And if the sides tilt rather than standing straight up, the flat miter is no longer correct. A canted box or a crown molding needs both a miter and a blade bevel.