Planters, tabletops, mirror frames, segmented turnings — anything with more than four sides. Give the number of sides and one dimension, and this gives you the miter angle, every side length, and how much stock to buy.
Notes
Why the miter is 180 divided by the sides
The angles of a closed polygon have to add up. With N sides you have N joints, each made of two mitered ends, so each cut is 180/N degrees. An octagon is 22.5, a hexagon is 30, a square is 45.
Across flats or across corners?
Across flats is wall to opposite wall — what fits in a space. Across corners is point to opposite point — what the piece measures at its widest. They are different numbers and mixing them up is the classic error.
Why it never closes perfectly
Half a degree of error per cut, doubled per joint, multiplied by eight joints, is four degrees of gap. Cut one test set in scrap, dry-assemble the whole ring, and adjust before touching good stock.
What this does not do
It assumes flat, untilted sides. Staved or tapered work — buckets, splayed planters — needs compound angles, which is a different calculation.