# 1.12: Polygons

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Define polygon.
• Identify polygons as convex or concave.
• Classify polygons by number of sides.
• Determine if a polygon is regular or not.

## Defining Polygons

A polygon is any closed planar figure that is made entirely of line segments that intersect at their endpoints. Polygons can have three or more sides and angles.

• A _________________________ is a closed figure made of straight line segments that intersect at their endpoints.
• Polygons can have __________________ or more sides and angles.

The segments are called the sides of the polygons, and the points where the segments intersect are called vertices. [Note that the singular of vertices is vertex.]

The following shape is a polygon:

• The segments in a polygon are called _____________________.
• Points where the sides of a polygon intersect are called _______________________.
• The plural form of vertex is ________________________________.

The prefix “poly” means “many.”

For example, the region of the world known as “Polynesia” is made of many islands.

Example 1

Which of the figures below is a polygon?

The easiest way to identify the polygon is to identify which shapes are not polygons.

Choices B and C each have at least one curved side. So they cannot be polygons because polygons must have straight sides.

Choice D has all straight sides, but one of the vertices is not at the endpoints of the two adjacent sides, so it is not a polygon.

Choice A is composed entirely of line segments that intersect at their endpoints. So, it is a polygon. The correct answer is A.

## Convex and Concave Polygons

Now that you know how to identify polygons, you can begin to practice classifying them.

The first type of classification to learn is whether a polygon is convex or concave. Think of the term concave as referring to a cave, or an interior space. A concave polygon has a section that “points inward” toward the middle of the shape. In any concave polygon, there are at least two vertices that can be connected without passing through the interior of the shape. The polygon below is concave and demonstrates this property:

• A ________________________ polygon has a section that points inward.

## Diagonals

A convex polygon does not share this property. Any time you connect the vertices of a convex polygon, the segments between non-adjacent vertices will travel through the interior of the shape.

• In a convex polygon, the vertices are only connected in the _____________________ of the shape.

Lines segments that connect to vertices traveling only on the interior of the shape are called diagonals.

• A diagonal is a line segment in a ______________________ polygon that connects the vertices only on the interior of the shape.

Identify each polygon below as concave or convex. If you do not think that the shape is a polygon, identify it as “not a polygon.”

__________________ __________________ _________________ ________________

## Classifying Polygons by their Numbers of Sides

The most common way to classify a polygon is by the number of sides. Regardless of whether the polygon is convex or concave, it can be named by the number of sides.

The prefix in each name reveals the number of sides. Refer to the polygon chart at the end of this lesson to name and classify samples of polygons.

## Regular, Equiangular and Equilateral Polygons

Polygons that have congruent sides are equilateral.

Polygons that have congruent angles are equiangular.

If a polygon is both equilateral and equiangular, it is called a regular polygon.

A square is an example of a regular polygon because it has four congruent angles and four congruent sides.

• Equilateral polygons have congruent _______________________.
• Equiangular polygons have congruent _______________________.
• A regular polygon is ______________________ and _______________________.

You can probably figure out that “equi” means “equal” and “angular” means “angles.”

But what about “lateral”? Lateral is related to the Spanish word “lado,” which means “side.”

What do you think the word “quadrilateral” means when you break it down?

“Regular” in this case means “consistent,” as in “the song has a regular rhythm” or “her regular bedtime is 10:00.”

Identify each polygon below as equiangular, equilateral, or regular.

______________________ ____________________ ______________________

## Graphic Organizer for Lesson 9

Classifying Polygons by their Numbers of Sides

The most common way to classify a polygon is by the number of sides. Regardless of whether the polygon is convex or concave, it can be named by the number of sides. The prefix in each name reveals the number of sides. The chart below shows names and samples of polygons.

Polygon Name Number of Sides Sample Drawings Draw your own example Can you think of any other words that begin with...
Triangle 3

...tri?

• Tricycle
• Triple

...qua?

• Quarter
Pentagon 5

...penta?

• Pentameter
Hexagon 6
Heptagon 7
Octagon 8

...octo?

• Octopus
Nonagon 9
Decagon 10

...deca?

Undecagon 11
Dodecagon 12
n\begin{align*}n-\end{align*}gon n\begin{align*}n\end{align*} (where n>3\begin{align*}n > 3\end{align*})

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