# Regular Polygons

In elementary geometry, a **polygon** is a plane figure that is bounded by a
finite chain of straight line segments closing in a loop to form a closed chain
or circuit. These segments are called its **edges** or **sides**, and the
points where two edges meet are the polygon's **vertices** (singular: **vertex**)
or **corners**. Polygons can be regular as in the **Table 1** below, List of regular polygons or
irregular.
For a typical polygon with **n** sides, the sum of angles = **(n – 2) * 180**.
This rule is derived from the fact that, any regular polygon can be divided into
**(n-2)** triangles by drawing diagonals from one corner to all other corners.
No two diagonals should cross, as in the diagram shown in Fig 1. The size of each
internal angle is therefore = **[(n – 2) * 180] /n**.

In a typical polygon such as a hexagon where the:

Total number of sides, **n = 6**

Total max number of triangles = **4** as in Fig 1. Total number of triangles
**4** is given by **(n – 2)**.

The sum of angles in a triangles = **180 °**.

Therefore, the sum of angles in a polygon = **(n - 2) * 180**.

The **area** for any regular polygon can be given in terms of the **side s**,
the **inscribed circle radius r**, also known as **apothem** or the **circumcircle
radius R**.

For a typical regular polygon such as the hexagon in the Fig 2, **s**, **r**
and **R** form two triangles **ABC** and **AEC**. Using these two triangles
in combination with trigonometric identities, and the internal angle **θ**, the
area of a regular polygon can be found. **α** is the external angle to the polygon,
it is also supplementary to **θ**.

The following is a step by step proof of the area of a regular polygon in terms
of **s**, **n** and **θ**.

The following are the other formulae for the area of the regular polygon.

Where

And

Two other formulae can be worked out in terms of **r** or **R** using the
same triangle **AEC** from Fig 3. These formulae are shown to the left of the above
table.

The centroid of a regular polygon is where all the diagonals meet. For irregular
polygon, visit the Irregular Polygons page.

### List of regular polygons

The table below lists some of the most popular regular polygons and some of their properties. To find out more about any of the polygons, and how to construct them and an in-depth details of their geometric properties, click on the name of the polygon. For the list of all polygons' names, visit this link List of Polygons. For other geometric shapes visit the Geometric Properties page.

#### Table 1

List of regular polygons (www.gimaths.com) | ||||||
---|---|---|---|---|---|---|

Geomentic Shape | Sum of Inerior Angles |
Interior
Angle |
Exterior
Angle |
Number of Triangles |
||

Name | Shape | Sides / Angles | ||||

Triangle Trigon |
3 | 180 ° | 60 ° | 120 ° | 1 | |

Quadrilateral Tetragon |
4 | 360 ° | 90 ° | 90 ° | 2 | |

Pentagon | 5 | 540 ° | 108 ° | 72 ° | 3 | |

Hexagon | 6 | 720 ° | 120 ° | 60 ° | 4 | |

Heptagon | 7 | 900 ° | 128.57 ° | 51.43 ° | 5 | |

Octagon | 8 | 1080 ° | 135 ° | 45 ° | 6 | |

Regular Polygon | n | (n - 2) * 180 | (n - 2) * 180/n | 180-[(n-2)*180/n] | (n - 2) |