Regular Polygon Calculator

What is a Regular Polygon? Definition, Angular Mechanics & Symmetry

In Euclidean plane geometry, a regular polygon is a closed two-dimensional convex figure that is both equilateral (all $n$ sides share identical length $s$) and equiangular (all $n$ interior angles share identical angular measure $\alpha$). Because its vertices are spaced evenly across a circular locus, every regular polygon possesses both a concentric incircle (inscribed circle tangent to all edges) and a circumcircle (circumscribed circle passing through every vertex).

A regular polygon has exactly two degrees of freedom: the integer count of its sides ($n \ge 3$) and any single linear dimension—such as side length ($s$), apothem ($a$), circumradius ($R$), perimeter ($P$), or enclosed surface area ($A$). Once the number of sides $n$ and any one of these linear parameters are established, all other angular, radial, and spatial dimensions are rigidly determined.

Core Mathematical Properties & Angular Mechanics

• Interior Angle Measure ($\alpha$): The sum of all interior angles across any $n$-sided polygon is strictly $(n - 2) \times 180^\circ$. For a regular polygon, each single interior angle equals:
  $$\alpha = \frac{(n - 2) \times 180^\circ}{n}$$
• Exterior & Central Angles ($\theta_e, \theta_c$): A full perimeter rotation traces $360^\circ$ ($2\pi\text{ rad}$). In every regular polygon, each central angle subtending an edge and each exterior turn angle are identical:
  $$\theta_c = \theta_e = \frac{360^\circ}{n}$$
• Total Internal Diagonals ($N_d$): The total count of distinct internal chords connecting non-adjacent vertices is:
  $$N_d = \frac{n(n - 3)}{2}$$
• Isoperimetric Limit: As the side count $n$ approaches infinity ($n \to \infty$), the perimeter of the polygon approaches the circumference of a circle ($2\pi R$), and the apothem approaches the circumradius ($a \to R$). A regular polygon represents the maximum enclosed surface area achievable by any $n$-gon of fixed perimeter.

How to Use This Regular Polygon Calculator?

This tool computes the comprehensive geometry of any regular $n$-gon from minimal inputs:

1. Step 1: Specify Side Count ($n$): Input the total number of sides ($n \ge 3$, e.g., 3 for Equilateral Triangle, 4 for Square, 5 for Pentagon, 6 for Hexagon, 8 for Octagon).
2. Step 2: Enter Any One Known Linear Dimension: Provide either the Side Length ($s$), Apothem / Inradius ($a$), Circumradius ($R$), Perimeter ($P$), or Enclosed Area ($A$).
3. Step 3: Instant Live Evaluation: The system simultaneously evaluates total surface area, perimeter boundary, interior/exterior angles, circumcircle/incircle radii, and diagonal counts.
4. Step 4: Reverse Transformations:
   Side from Perimeter: $$s = \frac{P}{n}$$

   Side from Apothem: $$s = 2a \tan\left(\frac{\pi}{n}\right)$$

   Side from Circumradius: $$s = 2R \sin\left(\frac{\pi}{n}\right)$$

Regular Polygon Formulas Reference Table

Worked Practical Scenarios (Step-by-Step)

Where Regular Polygons Appear in Engineering & Architecture

Common Traps & Measurement Pitfalls

• Confusing Apothem ($a$) with Circumradius ($R$): The apothem is the perpendicular distance to the midpoint of a side (incircle radius), while the circumradius is the distance to a vertex corner. Because the apothem is a leg of the internal right triangle and $R$ is the hypotenuse, $a < R$ is always true for any finite polygon.
• Mixing Degrees and Radians in Trigonometric Functions: Evaluating $\tan(\pi / n)$ requires calculators to be set to Radians, whereas evaluating $\tan(180^\circ / n)$ requires Degree mode.
• Assuming Diagonals Equal the Diameter: In regular polygons with an odd number of sides ($n = 5, 7, 9$), no diagonal passes through the center point; only even-sided polygons ($n = 4, 6, 8$) have principal diagonals that form true central diameters ($d = 2R$).