Polygon Calculator

Calculate properties of regular polygons such as area, perimeter, and angles.

Enter the number of sides and the length of one side to instantly compute all the key metrics of a regular polygon.

Examples

Click on an example to load its data into the calculator.

Equilateral Triangle

Regular Polygon

A polygon with 3 equal sides and angles.

Number of Sides: 3

Side Length: 10

Square

Regular Polygon

A polygon with 4 equal sides and right angles.

Number of Sides: 4

Side Length: 15

Regular Hexagon

Regular Polygon

A six-sided polygon, often seen in honeycombs.

Number of Sides: 6

Side Length: 8

Regular Octagon

Regular Polygon

An eight-sided polygon, famous for stop signs.

Number of Sides: 8

Side Length: 5

Other Titles
Understanding the Polygon Calculator: A Comprehensive Guide
Explore the geometry of regular polygons, from basic properties to the complex formulas that define them.

What is a Regular Polygon?

  • Defining the core characteristics of a regular polygon
  • Understanding the difference between regular and irregular polygons
  • Identifying key terminology like vertices, sides, and angles
In geometry, a polygon is a two-dimensional closed shape made of straight line segments. A regular polygon is a special type of polygon that is both equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Examples include the equilateral triangle, the square, the regular pentagon, and the regular hexagon.
In contrast, an irregular polygon does not have equal sides and angles. This calculator focuses exclusively on regular polygons, as their symmetry allows for straightforward calculation of their properties using simple formulas.
Key Terminology
Side (s): One of the line segments that make up the polygon. In a regular polygon, all sides are of equal length.
Vertex: A point where two sides meet. A polygon with 'n' sides also has 'n' vertices.
Interior Angle: The angle formed inside the polygon at a vertex. In a regular polygon, all interior angles are equal.
Exterior Angle: The angle formed outside the polygon by extending one of the sides. It is supplementary to the interior angle.

Common Regular Polygons

  • Equilateral Triangle (3 sides)
  • Square (4 sides)
  • Regular Pentagon (5 sides)
  • Regular Hexagon (6 sides)

Step-by-Step Guide to Using the Polygon Calculator

  • Entering the correct inputs for accurate calculations
  • Navigating the different input fields and options
  • Interpreting the calculated results effectively
Our Polygon Calculator is designed for ease of use. Follow these simple steps to calculate the properties of any regular polygon.
Input Instructions:
1. Number of Sides (n): Enter the number of sides your regular polygon has. This must be an integer of 3 or greater. The calculator will automatically identify the polygon type (e.g., entering '5' will be identified as a Pentagon).
2. Side Length (s): Input the length of a single side. Since the polygon is regular, all sides are the same length. This must be a positive number.
Calculation and Results:
Once you have entered both values, click the 'Calculate' button. The tool will instantly display the following results:
  • Area: The total space enclosed by the polygon.
  • Perimeter: The total length of the polygon's boundary.
  • Sum of Interior Angles: The sum of all angles inside the polygon, in degrees.
  • Interior and Exterior Angles: The measure of a single interior and exterior angle, in degrees.
  • Number of Diagonals: The total count of lines that can be drawn between non-adjacent vertices.

Example Walkthrough

  • Input: Number of Sides = 6, Side Length = 10
  • Result: Hexagon, Perimeter = 60, Area ≈ 259.81
  • Input: Number of Sides = 4, Side Length = 5
  • Result: Square, Perimeter = 20, Area = 25

Mathematical Formulas and Derivations

  • The formula for the area of a regular polygon
  • Calculating the perimeter and angles
  • Deriving the number of diagonals
The calculator uses standard geometric formulas to compute the properties of regular polygons. Here's a look at the mathematics behind the results.
Perimeter (P)
The perimeter is the easiest property to calculate. It's simply the number of sides (n) multiplied by the side length (s). Formula: P = n × s
Area (A)
The area of a regular polygon is more complex. The standard formula uses the number of sides, side length, and the tangent function. Formula: A = (s² n) / (4 tan(180°/n))
Angles
  • Sum of Interior Angles: This depends only on the number of sides. Formula: (n - 2) × 180°
  • Single Interior Angle: For a regular polygon, just divide the sum by the number of sides. Formula: ((n - 2) × 180°) / n
  • Single Exterior Angle: The sum of exterior angles is always 360°. Formula: 360° / n
Number of Diagonals
A diagonal is a line segment connecting two non-consecutive vertices. The formula to find the total number of diagonals is: D = n × (n - 3) / 2

Formula Application (for a Pentagon, n=5, s=10)

  • Perimeter = 5 × 10 = 50
  • Sum of Angles = (5 - 2) × 180° = 540°
  • Interior Angle = 540° / 5 = 108°
  • Diagonals = 5 × (5 - 3) / 2 = 5

Real-World Applications of Polygons

  • Polygons in architecture and construction
  • Their significance in art, design, and branding
  • Natural occurrences of polygonal shapes
Polygons are not just abstract geometric shapes; they are fundamental building blocks found all around us.
Architecture and Engineering
Hexagonal nuts and bolts provide excellent grip and torque distribution. Buildings often use square and rectangular footprints for stability and ease of construction. The Pentagon building in the United States is a world-famous example of polygonal architecture.
Nature
Nature is filled with polygonal shapes. Honeycombs, built by bees, are arrays of perfect hexagons, a shape that tessellates perfectly to store the most honey with the least amount of wax. Snowflakes exhibit intricate six-fold hexagonal symmetry. Columnar basalt formations, like the Giant's Causeway, often form as hexagonal columns of rock.
Art and Design
Artists and designers use polygons to create patterns, logos, and aesthetically pleasing compositions. Tiling (tessellation) with polygons is a common artistic technique. In digital graphics, complex surfaces are rendered using a mesh of tiny polygons (typically triangles), a technology central to video games and CGI.

Application Examples

  • Stop signs in many countries are regular octagons.
  • Floor tiles are often squares or hexagons.
  • Soccer balls are constructed from a combination of pentagons and hexagons.

Exploring Different Types of Polygons

  • A list of common polygons and their names
  • Unique properties of specific polygons
  • Understanding polygons with a large number of sides
As the number of sides increases, the name of the polygon changes. Here is a list of some common polygons.
List of Polygons by Number of Sides
  • 3 sides: Triangle
  • 4 sides: Quadrilateral (Square for regular)
  • 5 sides: Pentagon
  • 6 sides: Hexagon
  • 7 sides: Heptagon
  • 8 sides: Octagon
  • 9 sides: Nonagon
  • 10 sides: Decagon
  • 12 sides: Dodecagon
  • 20 sides: Icosagon
Approaching a Circle
An interesting property of regular polygons is that as the number of sides (n) approaches infinity, the polygon begins to look more and more like a circle. Its interior angle approaches 180°, and the ratio of its perimeter to the distance from the center to a vertex approaches 2π, just like in a circle.

Did You Know?

  • A chiliagon is a polygon with 1,000 sides.
  • A megagon has one million sides.
  • Apeirogon is a degenerate polygon with a countably infinite number of sides.