Polygons

What is a Polygon?

The word "polygon" comes from Greek.  "Poly" means "many," and "gon" has the meaning of "angle."  So a polygon is a figure with "many angles."  Another way to think of polygon is a figure with many sides, because angles have sides.  But this is not clear enough to distinguish what a polygon really is.  Some polygons you already know include triangles and rectangles.  In geometry we use definitions to classify objects that have certain properties in common and then state theorems about those properties.  For polygons those properties will relate to angles.

Recall that a good definition has to satisfy four conditions:

        1)    It names the term being defined.

        2)    It identifies the set to which the term belongs.

        3)    It states the properties that distinguish the term from others in the set.

        4)    It is reversible.

To properly define a polygon, we need to first understand what kinds of objects belong to the set of all polygons.  The main concerns in geometry are developing theorems that apply to all these objects.  In the next lesson we will state some theorems about the relationships between angles and sides of polygons.  But first let us consider some examples.  One condition is that polygons are 2-dimensional figures.  The question then is, which of those figures can properly be classified as polygons?

Definition of Polygon

In the following set of figures, only b, d, and e are polygons:

Notice that the polygons only contain (straight) segments, these segments do not overlap, and the endpoints of segments are always joined.  An important property of any polygon is that it can be completely filled with non-overlapping triangles with their vertices at the vertices of the polygon.

In view of this, we can define a polygon as follows:

DEFINITION:     A polygon is a two-dimensional figure consisting of at least 3 vertices (points) and at least 3 sides (straight segments) such that:

1)            each vertex is the endpoint of exactly 2 sides

2)            each pair of vertices are the endpoints of exactly one side

3)            if 2 sides intersect then their intersection is a vertex.

Now let's test this definition to see if it covers everything about polygons and nothing more.  First consider the three polygons:

The vertices are the endpoints of the sides.  In b there are 12 vertices, in d there are 10, and in e there are 4.  They are the "corners" where the sides meet.  Clearly these three figures satisfy the definition above.  (By the way, vertices is plural of vertex.)

Now consider the figures that are not polygons:

Figure a is not a polygon because vertices A and C are not the endpoints of the same side.  Figure c is not a polygon because vertex E is the endpoint of more than two sides (sides  .)  Figure f is not a polygon because side  is not a segment (straight).

Classifications of Polygons

You might be aware of certain names of special polygons, such as pentagon, hexagon, and octagon.  These terms classify a polygon by the number of sides.  In fact, the common names of polygons with 3 to 10 sides are: 

You may be surprised that an object like this is called a pentagon:
We are accustomed to seeing pentagons that look like this:

Both these figures are polygons with 5 sides according to our definition.  What is peculiar about the first figure is it "caves in" at the top.  Such a polygon is called concave.  In fact, a definition of "concave" is this:

DEFINITION:  A polygon is concave if there are two points somewhere inside it for which a segment with these as its endpoints cuts at least 2 of the sides of the polygon.

For example,

is concave, because the segment with points A and B as its endpoints cuts two sides:

A polygon that is not concave is called convex.  Some important theorems that apply only to convex polygons will be stated in the next lesson.

Regular Polygons

A special kind of convex polygon is called a regular polygon:

DEFINITION:    A regular polygon is a convex polygon in which all sides are congruent.

Another way to define a regular polygon is to state that its sides are congruent and its angles are congruent.  If a polygon is not regular, then it is called irregular.

Examples:

Concave Polygons:

Convex, Irregular Polygons:

Regular Polygons: