Regular Polygon Calculator
Given the number of sides plus any one of side length, apothem, circumradius, area or perimeter, get every other value — including interior and exterior angles, incircle and circumcircle areas — for any regular polygon (n ≥ 3). Live diagram, selected-polygons reference table and step-by-step working.
Material / cost estimator
Enter a price per square metre (tile, sheet metal, decking, land) to estimate total cost from the polygon's area.
What is a regular polygon?
A regular polygon is a closed 2D shape with n straight sides that are all the same length and n interior angles that are all equal. Equilateral triangles, squares, regular pentagons, hexagons and octagons are the familiar cases; the formulas below work for any integer n ≥ 3.
The shape is fully described by n plus any single length — the side s, apothem r, circumradius R, perimeter P or area A. Every other measurement is then determined.
Regular polygon, case by case
Given n and side s — the classic case
Slice the polygon into n identical isosceles triangles from the centre. Each has base s and height r (the apothem). Add them up and you get the standard area formula.
Given n and apothem r
The apothem (a.k.a. inradius) is the perpendicular distance from the centre to a side — the radius of the inscribed circle. Sides and area follow directly.
Given n and circumradius R
The circumradius is the centre-to-vertex distance — the radius of the circumscribed circle. Every regular polygon fits perfectly inside a circle of radius R.
Given n and area A — reverse-solve the side
Rearrange the area formula for s. Useful when you know how much surface a tile or panel should cover but need to work back to the edge length.
Given n and perimeter P
Because every side has the same length, the perimeter divides evenly: s = P/n. From there every other quantity follows.
Interior and exterior angles
The interior angle is the angle between two adjacent sides at any vertex. The exterior angle is its supplement — equivalently, the central angle subtended by one side. They always sum to 180°.
Formulas at a glance
All formulas — every calculation mode
The symbolic chain each mode of this calculator evaluates. n is the number of sides, s the side length, r the apothem (inradius), R the circumradius, P the perimeter and A the area.
Selected regular polygons
| n | Name | Shape | Interior x | Exterior y | Area (s = 1) |
|---|---|---|---|---|---|
| 3 | Equilateral triangle (trigon) | 60° | 120° | 0.43301 | |
| 4 | Square (tetragon) | 90° | 90° | 1 | |
| 5 | Pentagon | 108° | 72° | 1.72048 | |
| 6 | Hexagon | 120° | 60° | 2.59808 | |
| 7 | Heptagon | 128.5714° | 51.4286° | 3.63391 | |
| 8 | Octagon | 135° | 45° | 4.82843 | |
| 9 | Nonagon (enneagon) | 140° | 40° | 6.18182 | |
| 10 | Decagon | 144° | 36° | 7.69421 | |
| 11 | Hendecagon | 147.2727° | 32.7273° | 9.36564 | |
| 12 | Dodecagon | 150° | 30° | 11.19615 | |
| 13 | Tridecagon | 152.3077° | 27.6923° | 13.18577 | |
| 14 | Tetradecagon | 154.2857° | 25.7143° | 15.3345 | |
| 15 | Pentadecagon | 156° | 24° | 17.64236 | |
| 16 | Hexadecagon | 157.5° | 22.5° | 20.10936 | |
| 17 | Heptadecagon | 158.8235° | 21.1765° | 22.73549 | |
| 18 | Octadecagon | 160° | 20° | 25.52077 | |
| 19 | Enneadecagon | 161.0526° | 18.9474° | 28.46519 | |
| 20 | Icosagon | 162° | 18° | 31.56876 |
Interior angle x = (n−2)·180°/n and exterior y = 360°/n. The "Area (s = 1)" column equals ¼·n·cot(π/n) — the area of a regular n-gon whose side length is one unit; scale by s² for any other side.
Where the formulas come from
Slice from the centre. Draw straight lines from the centre to every vertex. You get n identical isosceles triangles, each with two sides equal to the circumradius R and an apex angle of 2π/n. The base of each triangle is one side s, and its height is the apothem r.
Side. Half of one triangle's base is the opposite leg of a right triangle with hypotenuse R and angle π/n: s/2 = R·sin(π/n). Similarly the adjacent leg is the apothem: r = R·cos(π/n). Rearranging gives the side ↔ apothem and side ↔ circumradius formulas.
Area. Each isosceles triangle has area ½·s·r, and there are n of them, so A = ½·n·s·r = ½·P·r. Substituting r = ½·s·cot(π/n) gives the closed form A = ¼·n·s²·cot(π/n).
Angles. Any convex n-gon has interior angles summing to (n − 2)·180°; dividing by n gives the interior angle of a regular polygon. The exterior angle is the supplement and equals the central angle each side subtends.
Worked example — hexagonal floor tile
A hexagonal tile has side s = 15 cm. What is its area, perimeter, and how large is the inscribed circle you could stamp into the middle of it?
So a 15 cm hexagon covers about 584.6 cm² and would just contain a circle of radius 12.99 cm (diameter ≈ 25.98 cm).
Worked example — octagonal gazebo floor
You want a regular-octagon gazebo floor with area A = 20 m². What side length and outer diameter do you need to cut to?
The centre-to-vertex measurement is about 2.66 m, so the octagon just fits inside a circle roughly 5.31 m across.
What this tool does for you
- Five solve modes — given n plus any one of side, apothem, circumradius, area or perimeter
- Handles every integer n ≥ 3 (up to 10 000), not just the named list — pick a preset or type a custom value
- Reports side, perimeter, apothem, circumradius, area, interior/exterior angles, and incircle/circumcircle areas
- Live SVG diagram labeled with s, r and R plus incircle and circumcircle overlays
- Unit selector (mm, cm, m, km, in, ft, yd) with automatic m² conversion for the cost estimator
- Significant-figures control (2–8) for all reported numbers
- Selected-polygons reference table from trigon (n = 3) to icosagon (n = 20) with interior and exterior angles and unit-side area
- Show/hide step-by-step working for every mode
- Material or cost estimator — enter a price per m² and get the total