Definition:Polygon

Definition

A polygon is comprised of an unspecified number of non-crossing straight line segments that connect in pairs at their endpoints (called vertices) to form a closed plane figure.

For example:

Parts of a Polygon

Side

The lines which make up the polygon are known as its sides.

Thus, in the diagram above, the sides are $a, b, c, d$ and $e$.

Vertex

The corners of a polygon are known as its vertices (singular: vertex).

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

Note that Euclid, in The Elements, uses the term vertex to mean the corner of a polygon furthest up the page from the base.

See height.

In the context of a triangle, the word apex should be used in preference.

Adjacent

Each vertex 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 end 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.

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 angle likewise has an opposite angle.

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

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

Internal Angle

The internal angle (or interior angle) of a vertex is the size of the angle between the sides forming that vertex, as measured inside the polygon.

External Angle

Surprisingly, the external angle (or exterior angle) of a vertex is not the size of the angle between the sides forming that vertex, as measured outside the polygon.

It 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 $\angle EFG$.

Note: it doesn't matter which adjacent side you use, since they are equal by the Vertical Angle Theorem.

Base

For a given polygon, one of the sides can be distinguished as being 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.

As Euclid defined it:

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

(The Elements: Book VI: Definition $4$)

Types of Polygon

Equilateral Polygon

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

The term is usually found in the term equilateral triangle.

Equiangular Polygon

An equiangular polygon is a polygon in which all the angles are the same.

Regular Polygon

A regular polygon is a polygon in which all the sides are the same length, and all the vertices have the same angle; that is, it is both equilateral and equiangular:

Triangle

A triangle (or rarely, trigon) is a polygon with exactly three sides.

Quadrilateral

A quadrilateral (or rarely, tetragon) is a polygon with exactly four sides.

Multi-lateral

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

As Euclid defined it:

Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multi-lateral those contained by more than four straight lines.

(The Elements: Book I: Definition $19$)

This definition is somewhat arbitrary and is rarely used, as its applications are limited.

There are specific names for polygons with specific numbers of sides, as follows:

• 5 sides: Pentagon
• 6 sides: Hexagon
• 7 sides: Heptagon
• 8 sides: Octagon
• 9 sides: Nonagon or Enneagon
• 10 sides: Decagon
• 11 sides: Hendecagon or Undecagon
• 12 sides: Dodecagon

The list goes on, but learning the names of them all is something which, mercifully, is rarely inflicted upon children nowadays.

Instead, the term $n$-gon is usually used nowadays to specify a polygon with a specific number, that is $n$, sides.

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.