Definition:Polygon

Definition
A polygon is a closed plane figure made up of an unspecified number of straight line segments, for example:


 * IrregularPolygon.png

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, 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.


 * InternalExternal.png

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.



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:


 * RegularPolygon.png

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.



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.