Inside corner and outside turn
The included inside corner is the angle between the two pieces inside a flat joint. A square corner is 90°. A regular hexagon has 120° inside corners. The outside turn is 180° minus that inside angle: 90° for the square and 60° for the hexagon.
For a symmetric joint with matching-width pieces, each cut shares half the outside turn. That gives 45° saw miters for a square and 30° saw miters for a hexagon.
Zero on a typical miter saw means square
The miter angle calculator uses zero for a square crosscut. A 30° setting is 30° away from square, not an included joint angle of 30°. The result gives a magnitude; the project still determines which end is cut left or right.
Do not assume that dividing an arbitrary inside angle by two gives a saw setting. That shortcut happens to agree for a 90° corner and fails for other angles.
Bevel adds a separate dimension
Miter changes the cut across the face of a piece lying flat. Bevel tilts the blade. A compound cut changes both. The flat-frame and segmented-ring tools here keep the bevel at zero. They do not solve crown molding, sloped staves or arbitrary three-dimensional joints.
A polygon has more than one diameter
A segmented ring’s diameter can be measured across flats or across corners. For a regular even-sided polygon, those are different physical spans. For an odd side count, twice the apothem is an inscribed-circle diameter rather than the span between parallel opposite flats.
The segmented ring calculator explicitly uses twice the apothem for its diameter input. It displays the larger across-corners diameter so the conventions can be compared. Match that definition before copying a dimension from another plan.
Confirm with a test assembly
Small setting errors accumulate around a frame or ring. Check the saw’s convention, verify its actual setting, and assemble a scrap test before committing the project stock. The geometry describes an ideal joint, not machine calibration or a validated structural assembly.