Free Regular Polygon Area Calculator — Supports n-sided Polygons

Regular Polygon Area Calculator — Fast & Accurate ResultsA regular polygon is a geometric figure with all sides equal in length and all interior angles equal. Common examples include the equilateral triangle (3 sides), the square (4 sides), and the regular pentagon (5 sides). Calculating the area of a regular polygon is straightforward once you know a couple of key measurements: the number of sides (n), the length of a side (s), and optionally the apothem (a) — the perpendicular distance from the center to a side. This article explains the relevant formulas, shows how a fast and accurate calculator works, offers worked examples, and points out practical considerations and edge cases.

Key formulas

There are several equivalent formulas to compute the area A of a regular n-sided polygon. Use whichever inputs you have:

• Using apothem (a) and perimeter (P): A = (⁄₂) · P · a
  where P = n · s.

• Using number of sides (n) and side length (s): A = (n · s^2) / (4 · tan(π / n))

• Using circumradius ® (distance from center to a vertex): A = (n / 2) · R^2 · sin(2π / n)

• Using apothem (a) and side length (s): A = (n · s · a) / 2

All formulas are mathematically equivalent; choose the one matching the inputs you have.

How a fast & accurate calculator works

A robust regular polygon area calculator follows these principles:

1. Input flexibility: Accepts combinations of (n, s), (n, a), (n, R), or (n, P). It validates that n is an integer ≥ 3 and that lengths are positive numbers.

2. Numeric stability: Uses well-conditioned formulas for small or large n. For large n where π/n is tiny, tan(π/n) ≈ π/n and direct evaluation of tan can introduce floating-point error; high-quality calculators switch to a series expansion or use built-in high-precision math libraries when n is large.

3. Unit consistency: Keeps units consistent and returns the area in squared units matching the input unit (for example, if s is in meters, area is in m^2).

4. Rounding and precision: Offers sensible default significant digits (e.g., 6–8 decimal places) and options for higher precision.

5. Error handling: Provides clear messages for invalid inputs (non-integer n, negative lengths, zero values) and edge cases.

Implementation examples

Below are concise algorithmic steps and a sample implementation in JavaScript (browser-friendly) and Python (widely used for scientific computation).

JavaScript:

// Regular polygon area from n (integer >=3) and side length s (>0) function regularPolygonAreaFromSide(n, s) {   if (!Number.isInteger(n) || n < 3) throw new Error("n must be integer >= 3");   if (s <= 0) throw new Error("side length must be > 0");   const angle = Math.PI / n;   return (n * s * s) / (4 * Math.tan(angle)); } 

Python:

import math def regular_polygon_area_from_side(n: int, s: float) -> float:     if not isinstance(n, int) or n < 3:         raise ValueError("n must be integer >= 3")     if s <= 0:         raise ValueError("side length must be > 0")     angle = math.pi / n     return (n * s * s) / (4 * math.tan(angle)) 

Worked examples

1. Square, n = 4, s = 5
   Using A = (n · s^2) / (4 · tan(π / n)): angle = π/4, tan(angle) = 1
   A = (4 · 25) / (4 · 1) = 25 → 25 square units

2. Regular hexagon, n = 6, s = 3
   angle = π/6, tan(π/6) = 1/√3 ≈ 0.577350269
   A = (6 · 9) / (4 · 0.577350269) ≈ 54 / 2.309401076 ≈ 23.3826859 → ≈23.3827 sq units

3. Regular 100-sided polygon, n = 100, s = 0.1
   angle = π/100 ≈ 0.0314159, tan(angle) ≈ 0.0314263
   A ≈ (100 · 0.01) / (4 · 0.0314263) ≈ 1 / 0.1257052 ≈ 7.9549 → ≈7.9549 sq units
   Note: for very large n, treating the polygon as a circle with radius ≈ s/(2 sin(π/n)) can be numerically stable.

Practical considerations

• If you have the apothem a and side s, computing area via A = (n · s · a) / 2 is straightforward and avoids trigonometric functions.
• If you only know the circumradius R, use A = (n / 2) · R^2 · sin(2π / n).
• For automated tools, support both decimal and common fraction inputs (e.g., “⁄₂”) and allow unit labels.
• Give users a diagram or brief explanation of apothem vs. circumradius to reduce input errors.

Edge cases and common mistakes

• Entering non-integer n: regular polygons require integer number of sides; a fraction like 4.5 is invalid.
• Zero or negative side lengths: physically meaningless and should trigger an error.
• Confusing apothem with circumradius: they differ; mixing them produces wrong results.
• Rounding too early: intermediate rounding can reduce accuracy, especially for large n.

Summary

A reliable “Regular Polygon Area Calculator — Fast & Accurate Results” accepts flexible inputs (n, s, a, R), validates values, uses numerically stable formulas, and presents results with clear units and controlled precision. For most users the formula A = (n · s^2) / (4 · tan(π/n)) is the most convenient when side length is known; if the apothem is known, use A = (⁄₂) · P · a for simplicity.