Hexagon from side length
For n = 6 and s = 5 cm: P = 6 × 5 = 30 cm; A = 6 × 5² / [4 tan(π/6)] = 64.9519 cm².
Regular Polygon Calculator: Area, Perimeter, Apothem & Angles
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Regular polygon formulas and guide
A regular polygon is a convex two-dimensional figure whose sides and interior angles are all equal. The apothem (or inradius) reaches the midpoint of a side at 90°, while the circumradius reaches a vertex.
Formulas by known measurement
| Known | Find side length | Find area |
|---|---|---|
| Side s | s | A = ns² / [4 tan(π/n)] |
| Apothem a | s = 2a tan(π/n) | A = na² tan(π/n) |
| Circumradius R | s = 2R sin(π/n) | A = nR² sin(2π/n) / 2 |
| Perimeter P | s = P/n | A = P² / [4n tan(π/n)] |
| Area A | s = √[4A tan(π/n)/n] | A |
| Perimeter and apothem | s = P/n | A = Pa/2 |
How to use the calculator
- Choose a familiar shape preset or enter a whole-number side count.
- Select what you know, enter a positive measurement, and choose its unit.
- Results update automatically. Adjust rounding or copy the results and formula.
Units, rounding, and accuracy
All length inputs must use the same unit. Perimeter, side, apothem, and radius use that unit; polygon and circle areas use the squared unit. The calculator keeps full JavaScript floating-point precision internally, then rounds only for display.
Apothem versus circumradius
Both start at the center, but the apothem ends at the midpoint of a side and the circumradius ends at a vertex. For a regular n-gon, a = R cos(π/n), so R is always at least as large as a.
Calculation methodology
The selected input is first converted to a side length. From that coherent value set, the calculator derives P = ns, a = s/[2 tan(π/n)], and R = s/[2 sin(π/n)]. Perimeter-plus-apothem mode uses the direct measured-area identity A = Pa/2.
Worked examples
Pentagon from apothem
For n = 5 and a = 4 m: s = 2 × 4 × tan(π/5) = 5.8123 m; A = 5 × 4² × tan(π/5) = 58.1234 m².
Octagon from circumradius
For n = 8 and R = 10 mm: s = 2 × 10 × sin(π/8) = 7.6537 mm; A = 8 × 10² × sin(π/4)/2 = 282.8427 mm².
Common regular polygons
| Polygon | Sides | Interior angle | Exterior angle | Diagonals |
|---|---|---|---|---|
| Equilateral triangle | 3 | 60° | 120° | 0 |
| Square | 4 | 90° | 90° | 2 |
| Pentagon | 5 | 108° | 72° | 5 |
| Hexagon | 6 | 120° | 60° | 9 |
| Octagon | 8 | 135° | 45° | 20 |
| Decagon | 10 | 144° | 36° | 35 |
| Dodecagon | 12 | 150° | 30° | 54 |
Editorial review and references
Author and calculation reviewer: Starlight Tools Mathematics Editorial Team
Last reviewed: 17 July 2026
Accuracy note: These formulas assume a convex regular polygon. Use consistent units. Displayed rounding may differ from the unrounded internal calculation, especially for very large or very small inputs.
References: OpenStax, Contemporary Mathematics 10.6: Area; Wolfram MathWorld: Regular Polygon; Carnegie Mellon University: formulas for angles of regular polygons.
Regular Polygon Calculator FAQs
What makes a polygon regular?
A regular polygon is convex and has equal side lengths and equal interior angles. Its vertices lie on one circumcircle, and every side is tangent to one incircle.
How do I calculate polygon area from a side length, apothem, perimeter, or circumradius?
For n sides, use A = ns²/[4 tan(π/n)] from side s, A = na² tan(π/n) from apothem a, A = Pa/2 from perimeter P and apothem a, or A = nR² sin(2π/n)/2 from circumradius R.
What is the difference between apothem and circumradius?
The apothem, or inradius, runs from the center perpendicular to a side. The circumradius runs from the center to a vertex, so it is longer. They satisfy a = R cos(π/n).
How many diagonals does a polygon have?
A polygon with n sides has n(n − 3)/2 diagonals. For example, a regular hexagon has 6(6 − 3)/2 = 9 diagonals.
Must the number of sides be a whole number?
Yes. A polygon must have a whole number of sides, and a convex polygon needs at least 3. This calculator accepts integers from 3 through 10,000.
How do units and rounding work?
Use one consistent length unit. Length results keep that unit and areas use its square. Calculations use full browser precision; Auto, decimal-place, significant-figure, and scientific display options only change the shown rounding.
Are my measurements uploaded?
No. The calculations and diagram run locally in your browser; measurement values are not uploaded by this calculator.