Definition:Polygon

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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:

IrregularPolygon.png

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.

InternalExternal.png

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 $P$:

ChordOfPolygon.png

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:

RegularPolygon.png


Triangle

Triangle.png

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.