Regular Polygon Calculator — Area, Perimeter, Side Length

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.

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.