# Polygon Calculator

## Calculate various properties of regular polygons

## Results

Property | Value |
---|---|

Side length (a) | – |

Inradius (r) | – |

Circumradius (R) | – |

Area (A) | – |

Perimeter (P) | – |

Interior angle | – |

Exterior angle | – |

Exploring Polygon Properties with a Polygon Properties Calculator

Polygons are fundamental geometric shapes with various properties and characteristics. They can range from simple polygons like triangles and squares to more complex ones like pentagons, hexagons, and beyond. Understanding the properties of polygons is essential in geometry and mathematics as a whole, as it enables us to analyze shapes, calculate areas, and solve real-world problems. A Polygon Properties Calculator is a valuable tool for exploring these shapes and their attributes.

# Polygon Calculator

This calculator helps you calculate properties of a regular polygon. By entering any one variable along with the number of sides or the polygon name, you can find side length, inradius (apothem), circumradius, area, and perimeter. The calculator works for regular polygons from a 3-gon up to a 1000-gon.

## Units

Note that units of length are shown for convenience and do not affect the calculations. They are used to give an indication of the order of the calculated results, such as ft, ft^{2}, or ft^{3}. Any other base unit can be substituted.

## Regular Polygon Formulas

A regular polygon is both equiangular and equilateral, meaning all sides are equal in length and all angles between sides are equal. When n = 3, it is an equilateral triangle, and when n = 4, it is a square.

The following formulas are used to calculate properties of a regular polygon where:

**a**= side length**r**= inradius (apothem)**R**= circumradius**A**= area**P**= perimeter**x**= interior angle**y**= exterior angle**n**= number of sides**π**= pi = 3.1415926535898**√**= square root

### Formulas

**Side Length (a)**: a = 2r tan(π/n) = 2R sin(π/n)**Inradius (r)**: r = (1/2)a cot(π/n) = R cos(π/n)**Circumradius (R)**: R = (1/2) a csc(π/n) = r sec(π/n)**Area (A)**: A = (1/4)na^{2}cot(π/n) = nr^{2}tan(π/n)**Perimeter (P)**: P = na**Interior Angle (x)**: x = ((n-2)π / n) radians = (((n-2)/n) x 180° ) degrees**Exterior Angle (y)**: y = (2π / n) radians = (360° / n) degrees

## Selected Polygons

Polygon Name | n | Polygon Shape | x | y |
---|---|---|---|---|

Trigon (Equilateral Triangle) | 3 | (1/3)π = 60° | (2/3)π = 120° | |

Tetragon (Square) | 4 | (2/4)π = 90° | (2/4)π = 90° | |

Pentagon | 5 | (3/5)π = 108° | (2/5)π = 72° | |

Hexagon | 6 | (4/6)π = 120° | (2/6)π = 60° | |

Heptagon | 7 | (5/7)π = 128.57° | (2/7)π = 51.43° | |

Octagon | 8 | (6/8)π = 135° | (2/8)π = 45° | |

Nonagon | 9 | (7/9)π = 140° | (2/9)π = 40° | |

Decagon | 10 | (8/10)π = 144° | (2/10)π = 36° | |

Undecagon | 11 | (9/11)π = 147.27° | (2/11)π = 32.73° | |

Dodecagon | 12 | (10/12)π = 150° | (2/12)π = 30° | |

Tridecagon | 13 | (11/13)π = 152.31° | (2/13)π = 27.69° | |

Tetradecagon | 14 | (12/14)π = 154.29° | (2/14)π = 25.71° |

## Diagram Examples

### Tetragon Diagram

A 4-sided polygon with **R** = circumradius, **r** = inradius (apothem), and **a** = side length.

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