Classifying Polygons

Closed shapes or figures in a plane with three or more sides are called polygons. Alternatively, a polygon can be defined as a closed planar figure that is the union of a finite number of line segments. In this definition, you consider closed as an undefined term. The term polygon is derived from a Greek word meaning “many‐angled.”

Polygons first fit into two general categories— convex and not convex (sometimes called concave). Figure 1 shows some convex polygons, some non‐convex polygons, and some figures that are not even classified as polygons.

Figure 1 Which are polygons? Which of the polygons are convex?

Identifying the parts of a polygon

The endpoints of the sides of polygons are called vertices. When naming a polygon, its vertices are named in consecutive order either clockwise or counterclockwise.

Consecutive sides are two sides that have an endpoint in common. The four‐sided polygon in Figure could have been named ABCD, BCDA, or ADCB, for example. It does not matter with which letter you begin as long as the vertices are named consecutively. Sides AB and BC are examples of consecutive sides.

Figure 2 There are four pairs of consecutive sides in this polygon.

A diagonal of a polygon is any segment that joins two nonconsecutive vertices. Figure 3 shows five‐sided polygon QRSTU. Segments QS , SU , UR , RT and QT are the diagonals in this polygon.

Figure 3 Diagonals of a polygon.

Number of sides

Polygons are also classified by how many sides (or angles) they have. The following lists the different types of polygons and the number of sides that they have:

  • A triangle is a three‐sided polygon.
  • A quadrilateral is a four‐sided polygon.
  • A pentagon is a five‐sided polygon.
  • A hexagon is a six‐sided polygon.
  • A septagon or heptagon is a seven‐sided polygon.
  • An octagon is an eight‐sided polygon.
  • A nonagon is a nine‐sided polygon.
  • A decagon is a ten‐sided polygon.

An earlier chapter showed that an equilateral triangle is automatically equiangular and that an equiangular triangle is automatically equilateral. This does not hold true for polygons in general, however. Figure shows examples of quadrilaterals that are equiangular but not equilateral, equilateral but not equiangular, and equiangular and equilateral.

Figure 4 An equiangular quadrilateral does not have to be equilateral, and an equilateral quadrilateral does not have to be equiangular.

Regular polygons

When a polygon is both equilateral and equiangular, it is referred to as a regular polygon. For a polygon to be regular, it must also be convex. Figure shows examples of regular polygons.

Figure 5 Regular polygons.

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