# Definition:Polygon

## Definition

A **polygon** is a closed plane figure made up of an unspecified number of non-crossing straight line segments that join in pairs at their endpoints.

For example:

## Parts of a Polygon

### Side

The line segments which make up a polygon are known as its **sides**.

Thus, in the polygon above, the **sides** are identified as $a, b, c, d$ and $e$.

### Vertex

A corner of a polygon is known as a **vertex**.

Thus, in the polygon above, the vertices are $A, B, C, D$ and $E$.

### Adjacent

Each vertex of a polygon is formed by the intersection of two sides.

The two sides that form a particular vertex are referred to as the **adjacents** of that vertex, or described as **adjacent to** that vertex.

Similarly, each side of a polygon intersects two other sides, and so is terminated at either endpoint by two vertices.

The two vertices that terminate a particular side are referred to as the **adjacents** of that side, or described as **adjacent to** that side.

Similarly, those two vertices are described as **adjacent to each other**.

Finally, two sides of a polygon that meet at the same vertex are **adjacent** to each other.

### Opposite

When a polygon has an even number of sides, each side has an **opposite side**, and each vertex likewise has an **opposite vertex**.

When a polygon has an odd number of sides, each side has an **opposite vertex**.

The **opposite side** (or **opposite vertex**) to a given side (or **vertex**) is that side (or vertex) which has the same number of sides between it and the side (or vertex) in question.

### Internal Angle

The **internal angle** of a vertex of a polygon is the size of the angle between the sides adjacent to that vertex, as measured *inside* the polygon.

### External Angle

Contrary to intuition, the **external angle** of a vertex of a polygon is *not* the size of the angle between the sides forming that vertex, as measured outside the polygon.

An **external angle** is in fact an angle formed by one side of a polygon and a line produced from an adjacent side.

While $\angle AFE$ is the internal angle of vertex $F$, the **external angle** of this vertex is $\angle EFG$.

### Base

For a given polygon, any one of its sides may be temporarily distinguished from the others, and referred to as the **base**.

It is immaterial which is so chosen.

The usual practice is that the polygon is drawn so that the **base** is made horizontal, and at the bottom.

### Height

The **height** of a polygon is the length of a perpendicular from the base to the vertex most distant from the base.

In the words of Euclid:

*The***height**of any figure is the perpendicular drawn from the vertex to the base.

(*The Elements*: Book $\text{VI}$: Definition $4$)

### Chord

A **chord** of a polygon $P$ is a straight line connecting two non-adjacent vertices of $P$:

In the above diagram, $DF$ is a **chord** of polygon $ABCDEFG$.

## Types of Polygon

### Equilateral Polygon

An **equilateral polygon** is a polygon in which all the sides are the same length.

### Equiangular Polygon

An **equiangular polygon** is a polygon in which all the vertices have the same angle.

### Regular Polygon

A **regular polygon** is a polygon which is both equilateral and equiangular.

That is, in which all the sides are the same length, and all the vertices have the same angle:

### Triangle

A **triangle** is a polygon with exactly three sides.

### Quadrilateral

A **quadrilateral** is a polygon with exactly four sides.

### Multi-lateral

A **multi-lateral polygon** is a term to define a polygon with more than four sides.

## Also see

- Results about
**polygons**can be found here.

## Note

The vertices and the sizes of the internal angles of those vertices are frequently referred to by the same letter.

Thus **the angle of vertex $A$** is called **angle $A$** and denoted $\angle A$.

This is considered by some to be an abuse of notation but its convenience outweighs its disadvantages.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**polygon**