Regular Polygon Calculator — Area, Perimeter, Side Length

Enter the number of sides n (n ≥ 3) and either side length or apothem. Private by design — runs locally in your browser.

Diagram & Inputs

Tip: Provide n (≥3) and either side or apothem. If both are given, we’ll cross-check them.

Results

How the Regular Polygon Calculator Works

This tool uses standard geometry relationships for regular n-gons. Once you provide the number of sides and either the side length s or the apothem a, all remaining properties follow directly:

  • Perimeter P = n·s
  • Apothem from side: a = s / (2·tan(π/n))
  • Side from apothem: s = 2·a·tan(π/n)
  • Area: A = ½·P·a = n·s² / (4·tan(π/n)) = n·a²·tan(π/n)
  • Circumradius: R = s / (2·sin(π/n)) = a / cos(π/n)
  • Angles: interior (n−2)·180°/n, exterior 360°/n, central 360°/n

All computation and rendering occur entirely in your browser.

5 Fun Facts about Regular Polygons

Hexagons are nature’s bargain

Honeybees picked the hexagon because it covers a plane with the least wall length for the same area—perfect for saving wax.

Efficiency champ

Pi by polygon

Archimedes boxed a circle between a 96‑gon and found 3.1408 < π < 3.1429. More sides, tighter bounds.

Ancient calculus

Tiling rule

Only three regular polygons tile the plane alone: equilateral triangles, squares, and regular hexagons. Everything else leaves gaps.

Symmetry trivia

Circle in disguise

As n grows, the apothem and circumradius converge, and the interior angle creeps toward 180°—a polygon smoothly morphs into a circle.

Limit intuition

Area loves halves

For any regular polygon, area = ½ · perimeter · apothem—the same structure as a triangle’s area, just multiplied by n matching wedges.

Formula echo

Understanding Regular Polygons: Definitions, Formulas, and Intuition

A regular polygon is a 2D shape with all sides equal and all angles equal. Triangles with three equal sides (equilateral triangles), squares, regular pentagons, and so on are all examples. Because of this symmetry, a single length plus the number of sides n determines almost everything you might want to know: perimeter, area, side length, apothem, circumradius, and angles. This calculator focuses on two practical inputs—side length (s) and apothem (a)—since these lead directly to compact, numerically stable formulas.

Key Quantities at a Glance

  • Perimeter: \( P = n \cdot s \)
  • Apothem (center to midpoint of a side): \( a = \dfrac{s}{2 \tan(\pi/n)} \)
  • Area (two equivalent forms): \( A = \tfrac{1}{2} P a = \dfrac{n s^2}{4 \tan(\pi/n)} = n a^2 \tan(\pi/n) \)
  • Circumradius (center to a vertex): \( R = \dfrac{s}{2 \sin(\pi/n)} = \dfrac{a}{\cos(\pi/n)} \)
  • Angles: interior \( = \dfrac{(n-2) \cdot 180^\circ}{n} \), exterior \( = \dfrac{360^\circ}{n} \), central \( = \dfrac{360^\circ}{n} \)

Where the Formulas Come From

Draw segments from the center to each vertex; the polygon splits into n congruent isosceles triangles with central angle \( \tfrac{2\pi}{n} \). Each base has length s, and the height of each triangle is the apothem a. The area of one triangle is \( \tfrac{1}{2} \cdot s \cdot a \); multiplying by n yields \( A = \tfrac{1}{2} P a \). Right-triangle trigonometry on half a side gives \( \tan(\pi/n) = \tfrac{s/2}{a} \Rightarrow a = \tfrac{s}{2 \tan(\pi/n)} \). Similarly, the circumradius comes from \( \sin(\pi/n) = \tfrac{s/2}{R} \Rightarrow R = \tfrac{s}{2\sin(\pi/n)} \). These relationships are exact for any regular n-gon.

Choosing an Input: Side vs. Apothem

If you know the side length, computing perimeter is immediate, and area follows from the apothem formula. If you know the apothem (common in tiling and machining), you can solve for the side with \( s = 2 a \tan(\pi/n) \). Supplying both lets you validate measurements; they must satisfy \( a = \tfrac{s}{2 \tan(\pi/n)} \). Small discrepancies typically indicate rounding or unit mismatches.

Units, Rounding, and Edge Cases

Perimeter and radii inherit the same linear units as your input (e.g., cm, in, m). Area uses squared units (e.g., cm²). For clean reports, round to a sensible number of decimals appropriate to your measurement precision. Remember that n must be an integer \( \ge 3 \). As n grows large, a regular polygon approaches a circle: \( a \approx R \) and the central angle \( \tfrac{360^\circ}{n} \) becomes very small, which is a useful intuition when approximating circular shapes with many sides.

Practical Applications

Regular polygons appear in architecture (window frames, domes), manufacturing (gear blanks, bolt circles), graphics and game design (meshes, sprites), and education (tilings, symmetry). Whether you’re checking a plan, estimating material, or teaching geometry, the side–apothem pair provides a fast, robust pathway to the quantities that matter.

Regular Polygon Calculator: FAQs

What inputs are required?

Enter the number of sides n (n ≥ 3) and either the side length or the apothem. If both are provided, the calculator cross-checks and warns if they disagree.

What is the apothem?

The apothem is the distance from the polygon’s center to the midpoint of a side, perpendicular to that side.

Which formulas does this use?

Perimeter P = n·s. Apothem a = s/(2·tan(π/n)). Area A = ½·P·a = n·s²/(4·tan(π/n)) = n·a²·tan(π/n). Circumradius R = s/(2·sin(π/n)) = a/cos(π/n).

Is my data private?

Yes. Everything runs locally; nothing is uploaded.

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