# Introduction to Polygons

You may want to review:

DEFINITION polygon
A polygon is a closed figure in a plane made by joining line segments, where each line segment intersects exactly two others.

Note: Strictly speaking, a polygon does not include its interior (the space inside the polygon).

A polygon
Not closed;
not a polygon
not a polygon
Line segment intersects more than two others;
not a polygon

Polygons are usually classified according to how many sides they have:

DEFINITIONS names of polygons with various numbers of sides
A triangle is a polygon with $\,3\,$ sides.
A quadrilateral is a polygon with $\,4\,$ sides.
A pentagon is a polygon with $\,5\,$ sides.
A hexagon is a polygon with $\,6\,$ sides.
A heptagon is a polygon with $\,7\,$ sides.
An octagon is a polygon with $\,8\,$ sides.
A nonagon is a polygon with $\,9\,$ sides.
A decagon is a polygon with $\,10\,$ sides.

More generally, a polygon with $\,n\,$ sides can be called an $\,n\,$-gon.

For example, a polygon with $\,27\,$ sides can be called a $\,27$-gon.

DEFINITION vertex of a polygon; vertices

The vertices of a polygon are the points where its sides intersect. The singular form of vertices is vertex.

## Naming Polygons

When naming polygons, the vertices must be listed in consecutive order. For example, the polygon above could be named:

• polygon $\,ABCD\,$ (start with $\,A\,,$ move clockwise)
• polygon $\,ADCB\,$ (start with $\,A\,,$ move counter-clockwise)
• polygon $\,BCDA\,$ (start with $\,B\,,$ move clockwise)
• polygon $\,BADC\,$ (start with $\,B\,,$ move counter-clockwise)
• polygon $\,CDAB\,$ (start with $\,C\,,$ move clockwise)
• polygon $\,CBAD\,$ (start with $\,C\,,$ move counter-clockwise)
• polygon $\,DABC\,$ (start with $\,D\,,$ move clockwise)
• polygon $\,DCBA\,$ (start with $\,D\,,$ move counter-clockwise)

More generally, when naming an $\,n$-gon, there are $\,n\,$ choices for listing the first vertex. Then, there are $\,2\,$ choices for the next vertex (moving clockwise or counterclockwise). The remaining vertices are then completely determined. Thus, there are $\,2n\,$ choices for the polygon name.

DEFINITION regular polygon
A regular polygon is a polygon whose sides all have the same length, and whose angles are all the same.
a regular triangle
(a square)
a regular pentagon
a regular hexagon
a regular heptagon
a regular octagon
a regular nonagon
a regular decagon
DEFINITIONS rectangle, square

A rectangle is a quadrilateral whose angles are all right angles.

A square is a rectangle with all sides of equal length.

Note: Every square is a rectangle. However, not every rectangle is a square. That is, there exist rectangles that are not squares.

Each of these is a rectangle, but not a square.

For fun, jump up to WolframAlpha and type in (say) ‘triangle’ or ‘quadrilateral’. You'll get loads of information!