Classifying Polygons
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C ONCEPT
1
1
Classifying Polygons
Learning Objectives
•
•
•
•
Define polygons.
Understand the difference between convex and concave polygons.
Classify polygons by number of sides.
Use the distance formula to find side lengths on a coordinate grid.
Introduction
As you progress in your studies of geometry, you can examine different types of shapes. In the last lesson, you studied the triangle, and different ways to classify triangles. This lesson presents other shapes, called polygons. There
are many different ways to classify and analyze these shapes. Practice these classification procedures frequently and
they will get easier and easier.
Defining Polygons
Now that you know what a triangle is, you can learn about other types of shapes. Triangles belong to a larger group
of shapes called polygons. A polygon is any closed planar figure that is made entirely of line segments that intersect
at their endpoints. Polygons can have any number of sides and angles, but the sides can never be curved.
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 easiest way to identify a polygon is to look for a closed figure with no curved sides. If there is any curvature in
a shape, it cannot be a polygon. Also, the points of a polygon must all lie within the same plane (or it wouldn’t be
two-dimensional).
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. 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.
Concept 1. Classifying Polygons
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Example 2
Which of the figures below is not a polygon?
All four of the shapes are composed of line segments, so you cannot eliminate any choices based on that criteria
alone. Notice that choices A, B, and D have points that all lie within the same plane. Choice C is a three-dimensional
shape, so it does not lie within one plane. So it is not a polygon. The correct answer is C.
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 convex polygon does not share this property. Any time you connect the vertices of a convex polygon, the segments
between nonadjacent vertices will travel through the interior of the shape. Lines segments that connect to vertices
traveling only on the interior of the shape are called diagonals.
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Example 3
Identify whether the shapes below are convex or concave.
To solve this problem, connect the vertices to see if the segments pass through the interior or exterior of the shape.
A. The segments go through the interior.
Therefore, the polygon is convex.
B. The segments go through the exterior.
Therefore, the polygon is concave.
C. One of the segments goes through the exterior.
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Thus, the polygon is concave.
Classifying Polygons
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.
TABLE 1.1:
Polygon Name
Triangle
Number of Sides
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Sample Drawings
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TABLE 1.1: (continued)
Polygon Name
Heptagon
Number of Sides
7
Octagon
8
Nonagon
9
Decagon
10
Undecagon or hendecagon (there
is some debate!)
11
Dodecagon
12
Sample Drawings
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TABLE 1.1: (continued)
Polygon Name
n−gon
Number of Sides
n (where n > 12)
Sample Drawings
Practice using these polygon names with the appropriate prefixes. The more you practice, the more you will
remember.
Example 4
Name the three polygons below by their number of sides.
A. This shape has seven sides, so it is a heptagon.
B. This shape has five sides, so it is a pentagon.
C. This shape has ten sides, so it is a decagon.
Using the Distance Formula on Polygons
You can use the distance formula to find the lengths of sides of polygons if they are on a coordinate grid. Remember
to carefully assign the values to the variables to ensure accuracy. Recall from algebra that you can find the distance
between points (x1 , y1 ) and (x2 , y2 ) using the following formula.
q
Distance = (x2 − x1 )2 + (y2 − y1 )2
Example 5
A quadrilateral has been drawn on the coordinate grid below.
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What is the length of segment BC?
Use the distance formula to solve this problem. The endpoints of BC are (−3, 9) and (4, 1). Substitute −3 for x1 , 9
for y1 , 4 for x2 , and 1 for y2 . Then we have:
D=
D=
D=
D=
D=
So the distance between points B and C is
q
(x2 − x1 )2 + (y2 − y1 )2
q
(4 − (−3))2 + (1 − 9)2
q
(7)2 + (−8)2
√
49 + 64
√
113
√
113 , or about 10.63 units.
Lesson Summary
In this lesson, we explored polygons. Specifically, we have learned:
•
•
•
•
How to define polygons.
How to understand the difference between convex and concave polygons.
How to classify polygons by number of sides.
How to use the distance formula to find side lengths on a coordinate grid.
Polygons are important geometric shapes, and there are many different types of questions that involve them. Polygons are important aspects of architecture and design and appear constantly in nature. Notice the polygons you
see every day when you look at buildings, chopped vegetables, and even bookshelves. Make sure you practice the
classifications of different polygons so that you can name them easily.
Review Questions
For exercises 1-5, name each polygon in as much detail as possible.
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1.
2.
3.
4.
5.
6. Explain why the following figures are NOT polygons:
7. How many diagonals can you draw from one vertex of a pentagon? Draw a sketch of your answer.
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8. How many diagonals can you draw from one vertex of an octagon? Draw a sketch of your answer.
9. How many diagonals can you draw from one vertex of a dodecagon?
10. Use your answers to 7, 8, and 9 and try more examples if necessary to answer the question: How many
diagonals can you draw from one vertex of an n−gon?
Review Answers
1. This is a convex pentagon.
2. Concave octagon.
3. Concave 17−gon (note that the number of sides is equal to the number of vertices, so it may be easier to count
the points [vertices] instead of the sides).
4. Concave decagon.
5. Convex quadrilateral.
6. A is not a polygon since the two sides do not meet at a vertex; B is not a polygon since one side is curved; C
is not a polygon since it is not enclosed.
7. The answer is 2.
8. The answer is 5.
9. A dodecagon has twelve sides, so you can draw nine diagonals from one vertex.
10. Use this table to answer question 10.
TABLE 1.2:
Sides
3
Diagonals from One Vertex
0
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TABLE 1.2: (continued)
Sides
4
5
6
7
8
9
10
11
12
...
n
Diagonals from One Vertex
1
2
3
4
5
6
7
8
9
...
n−3
To see the pattern, try adding a “process” column that takes you from the left column to the right side.
TABLE 1.3:
Sides
3
4
5
6
7
8
...
n
Process
(3) − 3 = 0
(4) − 3 = 1
(5) − 3 = 2
(6) − 3 = 3
(7) − 3 = 4
(8) − 3 = 5
(n) − 3 =
Diagonals from One Vertex
0
1
2
3
4
5
...
n−3
Notice that we subtract 3 from each number on the left to arrive at the number in the right column. So, if the number
in the left column is n (standing for some unknown number), then the number in the right column is n − 3.
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Concept 1. Classifying Polygons