This polygon solver calculates the properties of regular polygons — interior and exterior angles, side length, the across-flats and across-corners sizes, the apothem, and the area — from whatever you know. It is invaluable for laying out hexagonal and octagonal work, columns, planters, and segmented rings.
Angles and key dimensions
A regular polygon with N sides has interior angles of (N−2) × 180 ÷ N degrees and exterior angles of 360 ÷ N. From the number of sides plus one size — side length, the across-flats (width between parallel faces), or the across-corners (point to point) — every other dimension follows by trigonometry. The solver fills them all in, including the apothem (centre to the middle of a side), which is what you need to set out the shape or cut its faces.
For makers, the across-flats and across-corners figures are the practical ones: across-flats sets the size of a hexagonal nut seat or an octagonal column face-to-face, while across-corners gives the diameter of the stock you need to start from.
Building polygonal work
To make an N-sided frame or column, each joint is cut at half the exterior angle. The solver's angles feed straight into mitre and segment setups, and its side lengths tell you how much material each face takes — connecting the geometry to the actual cuts on the saw.
Laying out a regular hexagon (6 sides) with 100 mm sides.
- Interior angle = (6−2) × 180 ÷ 6 = 120°.
- Across flats = side × √3 = 100 × 1.732 ≈ 173 mm.
- Across corners = 2 × side = 200 mm.
The hexagon has 120° corners, measures ~173 mm across flats and 200 mm across corners.
