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 $E$.
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 $ABCDEFG$.
Adjacent
Adjacent Side to Vertex
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.
Adjacent Sides
Two sides of a polygon that meet at the same vertex are adjacent to each other.
Adjacent Vertex to Side
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.
Adjacent Vertices
Those two vertices are described as 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.
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.
Notation and Terminology
Both the vertices and the sizes of the internal angles of those vertices are frequently referred to by the same symbol.
Thus the size of the internal angle at 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.
Also see
- Results about polygons can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): polygon
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): polygon
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): polygon