How to use the Regular Polygon Calculator
- Enter the number of sides — 3 for a triangle, 6 for a hexagon, and so on.
- Enter the length of one side; in a regular polygon all sides are equal.
- Read the area for material quantities.
- Use the apothem when you need the centre-to-edge distance, for instance to size a spanner or a socket.
- Use the interior angle when setting mitre cuts.
How the calculation works
A regular polygon divides cleanly into n identical isosceles triangles meeting at the centre. Each has the side length as its base and the apothem — the perpendicular distance from the centre to a side's midpoint — as its height. Summing them gives area as half the perimeter times the apothem, which is the polygon analogue of the circle's area formula, and becomes exactly that formula as n grows large.
The apothem itself comes from trigonometry. The central angle of each triangle is 360/n degrees, so half of it is 180/n, and the half-side over the apothem is the tangent of that half-angle. Rearranging gives the apothem as the side divided by twice the tangent of pi/n. Everything else — area, circumradius, angles — follows from that one relationship.
The interior angle, (n − 2) × 180 / n, is the number joiners and metalworkers actually need. A hexagon's 120° interior angle means each mitre is cut at 30° off square; an octagon's 135° means 22.5° cuts. Getting this wrong by even half a degree accumulates around the full circuit, which is why the last joint of a hand-cut polygon frame is always the one that will not close.
apothem = s / (2·tan(π/n)); A = (n·s·apothem)/2; interior angle = (n−2)·180/nSource: Standard regular-polygon relations; see Coxeter, Introduction to Geometry, Chapter 2.
Worked example
A hexagonal paver has 200 mm sides. What area does each cover, and what mitre angle joins them?
- Apothem: 200 / (2 × tan(30°)) = 200 / 1.1547 = 173.2 mm.
- Perimeter: 6 × 200 = 1,200 mm.
- Area: (1,200 × 173.2) / 2 = 103,920 mm² = 0.1039 m².
- Interior angle: (6 − 2) × 180 / 6 = 120°.
0.1039 m² per paver — about 9.6 pavers per square metre before cuts — with 120° interior angles.
Frequently asked questions
Does this work for irregular polygons?+
No. Irregular shapes need the shoelace formula applied to vertex coordinates, or division into triangles.
What is the difference between apothem and radius?+
The apothem reaches the midpoint of a side; the circumradius reaches a vertex. The circumradius is always the larger of the two.
Why does a polygon approach a circle as sides increase?+
The apothem converges on the radius and the perimeter on the circumference, so half-perimeter times apothem converges on πr².
How do I set a mitre saw for a polygon frame?+
Set it to half the exterior angle, which is 180/n degrees. For a hexagon that is 30°, for an octagon 22.5°.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.