Mathematics, Statistics & Geometry

Area of Regular Polygon Calculator

Calculate the area, perimeter, and apothem of any regular polygon. Supports any number of sides with a full geometric breakdown.

Area
172.048
Perimeter50
Interior Angle108°
Apothem (Inradius)6.882
Calculation StepsSides (n) = 5 Side Length (s) = 10 Area = (n * s²) / (4 * tan(π/n)) Area = (5 * 10²) / (4 * tan(π/5)) Area = 172.047740 Apothem = s / (2 * tan(π/n)) = 6.881910

Calculated locally in your browser. Fast, secure, and private.

Beyond Triangles and Squares

The Area of Regular Polygon calculator is a sophisticated tool for solving complex geometry problems. Whether you are working with a hexagon, octagon, or a polygon with 50 sides, this tool handles the trigonometric heavy lifting to provide area, perimeter, and apothem measurements.

A=n⋅s24⋅tan⁡(π/n)\begin{aligned} A = \frac{n \cdot s^2}{4 \cdot \tan(\pi/n)} \end{aligned}

Where:
A=
The surface area of the regular polygon
n=
The number of equal-length sides (must be ≥ 3)
s=
The length of any single side

Common Regular Polygons

  • 3 Sides: Equilateral Triangle.
  • 4 Sides: Square.
  • 5 Sides: Regular Pentagon.
  • 6 Sides: Regular Hexagon (common in nature and engineering).
  • 8 Sides: Regular Octagon (the shape of a stop sign).

The Role of the Apothem

The apothem (a) is crucial because the area can also be expressed as Area = 0.5 × Perimeter × Apothem. This relationship shows that any regular polygon can be viewed as a collection of n identical triangles meeting at the center.

Real-World Uses

  • Architecture: Designing gazebos, columns, and decorative flooring patterns.
  • Mechanical Engineering: Sizing nuts, bolts, and gears (many are hexagonal).
  • Game Design: Creating hex-grids for strategy games and board games.
  • Biology: Analyzing the structure of honeycombs and certain molecular patterns.

Frequently Asked Questions

A regular polygon is a two-dimensional shape where all sides have the same length and all interior angles are equal (e.g., equilateral triangle, square, regular pentagon).

The general formula is (n × s²) / (4 × tan(π/n)), where n is the number of sides and s is the side length.

The apothem is the distance from the center of a regular polygon to the midpoint of one of its sides. It is also the radius of the largest circle that can fit inside the polygon.

The formula for the interior angle of a regular polygon is ((n - 2) × 180) / n.

Yes, our calculator supports polygons with 3 or more sides, including hexagons, octagons, and beyond.

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