Free Regular Polygon Calculator

The Regular Polygon Calculator is a comprehensive tool for computing essential geometric properties of any regular polygon — from an equilateral triangle to a polygon with thousands of sides. Whether you need the area of a heptagon, the interior angles of a decagon, or the apothem of a 15‑gon, this calculator delivers instant results after entering a single known parameter. The following sections provide a definition of polygons, explain the characteristics of regular polygons, present a naming reference table, list the underlying formulas, and walk through a practical example.

Defining Polygons

A polygon is a closed, two‑dimensional figure whose sides are straight line segments. Familiar polygons include triangles, squares, pentagons, hexagons, and octagons. Shapes such as circles and ellipses are not polygons. Polygons can be classified by the number of sides, convexity (convex vs. concave), and symmetry. This tool concentrates on regular polygons — those that are both equilateral (all sides equal) and equiangular (all interior angles equal). Every regular polygon with n sides can be divided into n congruent isosceles triangles whose apexes meet at the polygon’s center.

Polygon Names by Side Count

The name of a polygon usually reveals its number of sides through Greek prefixes. The table below lists many regular polygons, giving their interior and exterior angles in degrees. For any n‑sided regular polygon, the interior angle α and exterior angle β follow these formulas:

$$\alpha = \frac{(n-2) \times 180^\circ}{n}, \qquad \beta = \frac{360^\circ}{n}$$

In radians, $\alpha = \dfrac{(n-2)\pi}{n}$ and $\beta = \dfrac{2\pi}{n}$.

Polygon Name	Sides (n)	Interior Angle α (°)	Exterior Angle β (°)
Triangle (equilateral)	3	60	120
Square (quadrilateral)	4	90	90
Pentagon	5	108	72
Hexagon	6	120	60
Heptagon	7	≈128.571	≈51.429
Octagon	8	135	45
Nonagon	9	140	40
Decagon	10	144	36
Hendecagon (undecagon)	11	≈147.273	≈32.727
Dodecagon	12	150	30
Triskaidecagon	13	≈152.308	≈27.692
Tetrakaidecagon	14	≈154.286	≈25.714
Pentadecagon	15	156	24
Hexakaidecagon	16	157.5	22.5
Heptadecagon	17	≈158.824	≈21.176
Octakaidecagon	18	160	20
Enneadecagon	19	≈161.053	≈18.947
Icosagon	20	162	18
Triacontagon	30	168	12
Tetracontagon	40	171	9
Pentacontagon	50	172.8	7.2
Hectagon (100-gon)	100	176.4	3.6
Chiliagon (1000-gon)	1000	179.64	0.36
Myriagon (10,000-gon)	10,000	179.964	0.036
Megagon (1,000,000-gon)	1,000,000	≈180	≈0

For polygons with 11 or more sides, it is often simpler to use the “n‑gon” notation (e.g., 11‑gon, 14‑gon, 100‑gon).

Key Formulas for Regular Polygons

Let n be the number of sides and a the side length. All remaining parameters can be derived from these equations:

• Perimeter: $P = n \cdot a$
• Area: $A = \dfrac{n \cdot a^{2} \cdot \cot\!\left(\dfrac{\pi}{n}\right)}{4}$ Other forms using circumradius $R$ or apothem $r$ also exist.
• Interior angle: $\alpha = \dfrac{(n-2)\pi}{n}$ rad $= \dfrac{(n-2)\cdot 180^\circ}{n}$
• Exterior angle: $\beta = \dfrac{2\pi}{n}$ rad $= \dfrac{360^\circ}{n}$
• Apothem (incircle radius): $r = \dfrac{a}{2 \tan\!\left(\dfrac{\pi}{n}\right)}$
• Circumradius: $R = \dfrac{a}{2 \sin\!\left(\dfrac{\pi}{n}\right)}$

These formulas are the foundation of the calculator, enabling it to function as a Regular Polygon Area Calculator, a Polygon Angle Calculator, a Regular Polygon Side Calculator, and a Regular Polygon Apothem & Circumradius Calculator — all through a single input.

Example Usage

Consider a regular nonagon (9 sides) with a known perimeter of 18 inches. To discover all its properties using the tool:

1. Enter 9 as the number of sides.
2. Input the perimeter (18 in) as the given value.
3. The calculator immediately returns:
   • Side length $a = 2$ in
   • Area $A \approx 24.727$ in²
   • Interior angle $\alpha = 140^\circ$
   • Exterior angle $\beta = 40^\circ$
   • Circumradius $R \approx 2.924$ in
   • Apothem $r \approx 2.748$ in

This example shows how one known quantity leads to a complete set of geometric measures, simplifying polygon analysis.

Summary

The Regular Polygon Calculator provides a fast, accurate way to determine side length, area, perimeter, angles, apothem, and circumradius for any regular polygon. It supports an arbitrary number of sides — from 3 to well beyond 1000 — and relies on the standard formulas derived above. Whether you are a student verifying homework, an architect designing a structure, or a hobbyist exploring geometry, this tool makes regular polygon calculations effortless.