# Polygon

A polygon is a closed plane figure formed by three or more line segments. The figure below shows a few polygon examples.

Some real life examples of polygons include the Pentagon, the headquarters of the United States Department of Defense, the Pyramids of Egypt, the shape of a stop sign (octagon), and much more.

## What is a polygon

A polygon is a two-dimensional (2D), closed, plane figure formed by a minimum of three line segments.

The line segments that form a polygon are called sides. Two connected sides form an angle at a point called a vertex. A diagonal is a line segment joining two non-consecutive vertices. In the polygon below,

, , , and are four sides. They form four angles: ∠A, ∠B, ∠C, and ∠D. and are two diagonals.### Polygon definition

A polygon is defined as a 2D plane figure that is made up of 3 or more connected line segments that form a closed shape. The minimum number of sides a polygon can have is three. Otherwise, a closed figure cannot be formed. Also, any curved closed figures are not polygons. All parts of a polygon must be straight.

Polygons are named based on the number of sides in the polygon. Generally, the name of a polygon ends in "-gon" which is derived from the Greek "gonia" meaning corner or angle. The first part of the name usually indicates the number of sides that the polygon has. Examples include pentagon (5), hexagon (6), octagon (8), and decagon(10).

### Polygon properties

Below are some properties of all polygons:

- All polygons are two-dimensional (2D).
- All polygons are made up of straight sides.
- The sum of the exterior angles of a polygon is always equal to 360°.
- If at least one of the interior angles of a polygon is greater than 180°, the polygon is a concave polygon.
- If all the interior angles of a polygon measure less than 180°, the polygon is a convex polygon.

## Polygon types

The various types of polygons are described below.

### Regular polygon

A regular polygon is a polygon in which all sides have equal length (equilateral) and all angles have equal measure (equiangular). Below are some regular polygon examples.

### Irregular polygon

An irregular polygon has at least one side or angle that is not equal. The figure below shows some irregular polygon examples.

Note that the 5-, 6-, and 8-gon could also have been called a pentagon, hexagon, and octagon respectively. We named them as we did just to show that all polygons can be named as n-gon, where n is the number of sides.

### Concave polygon

A concave polygon is a polygon in which at least one interior angle measures greater than 180°. A concave polygon is always an irregular polygon. The following figure shows few concave polygon examples. The interior angles larger than 180° are marked with a red arc.

### Convex polygon

A convex polygon is one in which all interior angles measure less than 180°. Note that a regular polygon is always convex. The figure below shows some convex polygon examples.

### Simple polygon

A simple polygon is one that does not intersect itself and has no holes. All of the above examples on this page are simple polygons.

### Complex polygon

A complex polygon is a polygon with self-intersecting sides. The figure below shows some complex polygon examples.

### Number of sides of a polygon

Polygons are also categorized based on the number of sides

Polygons are commonly classified based on the number of sides they have. In general, a polygon with n-number of sides is called an n-gon. Some important polygons have specific names, such as triangles, pentagons, hexagons, etc. The following are some examples.

Polygon | # of sides | Shape |
---|---|---|

Triangle | 3 | |

Quadrilateral | 4 | |

Pentagon | 5 | |

Hexagon | 6 | |

Octagon | 8 |

There are many other polygons, and each polygon above can be further classified. For example, a triangle can be further classified as an acute, obtuse, or right triangle. Learn more about these polygons by navigating this website.

## Polygon formulas

Below are a number of general formulas for finding the area, perimeter, and measures of the interior and exterior angles of a polygon.

### Area of a polygon

The formula for the area of a regular polygon is,

where a is the length of the apothem and n is the number of sides of the polygon.

### Polygon perimeter

The formula for the perimeter of a polygon is,

where n is the number of sides of the polygon and s is the length of each side.

### Sum of interior angles of a polygon

The formula for the sum of the interior angles of a regular polygon is,

where n is the number of sides of a regular polygon.

### Interior angle of a regular polygon

The formula for the measure of each interior angle of a regular polygon is,

where n is the number of sides of a regular polygon.

### Exterior angle of a regular polygon

The formula for the measure of each exterior angle of a regular polygon is,

where n is the number of sides of a regular polygon. Note that the sum of the exterior angles of any polygon is always 360°.