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Polygons are two-dimensional geometric figures with these characteristics:
These shapes are polygons:
These shapes are not polygons:
Look at the shapes above that are not polygons. Explain why each of these shapes does not fit the definition of a polygon.
Polygons can be classified according to the number of sides they have. Note 2
Name |
# of Sides |
Examples |
Triangle | 3 | |
Quadrilateral | 4 | |
Pentagon | 5 | |
Hexagon | 6 | |
Heptagon | 7 | |
Octagon | 8 | |
Nonagon | 9 | |
Decagon | 10 |
Polygons with more than 10 sides are not usually given special names. A polygon with 11 sides is described as an 11-gon, a polygon with 12 sides as a 12-gon, and so on. Each of the polygons below is a 17-gon.
When people talk about a general polygon — one where you don’t know the exact number of sides—they often refer to it as an n-gon.
Hidden Polygons and the Video Segment problem adapted from IMPACT Mathematics, developed by Educational Development Center, Inc. pp. 42-45. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math
Each corner of a polygon, where two sides meet, is called a vertex. The plural of vertex is vertices. Labeling vertices with capital letters makes it easy to refer to a polygon by name. For example, this figure contains two triangles and one quadrilateral:
To name one of the polygons in the figure, list its vertices in order as you move around it in either direction. One name for the shaded triangle is Triangle ABC. Other names are possible, including BCA and ACB. One name for the white triangle is Triangle ADC.
The quadrilateral in the figure could be named Quadrilateral ABCD, BCDA, DCBA, or DABC. All of these names list the vertices in order as you move around the quadrilateral. The name ACBD is not correct.
In the following activities, you will search for polygons in several figures. You’ll calculate a score for each figure by adding the following:
Be careful to give only one name for each polygon. You may want to record your work for each problem in a table like this one, which shows the result for this figure.As you work, try to discover a systematic way to find and list all the polygons in a figure.
Polygon |
Names |
Score |
Triangle | ABC, ADC | 6 |
Quadrilateral | ABCD | 4 |
Pentagon | None | – |
Hexagon | None | – |
Total Score | 10 |
Tip: Here’s a sample strategy for counting: Count triangles and quadrilaterals, and then look for their “complements.” So YXWZV is everything in the shape except triangle YVZ. Another strategy would be to choose one vertex (e.g., X). Count all of the triangles that contain X. Then count all of the quadrilaterals that contain X, and so on. Next, count all of the triangles that contain vertex Y but not X, and so on.
How many polygons can you find in the following figure?
How many polygons can you find in the following figure?
How many polygons can you find in the following figure?
Note 2
Discuss or reflect on why there is no name for a two-sided polygon. Namely, if two segments meet only at endpoints, the figure cannot be closed; therefore it cannot be a polygon.
The first two shapes are not polygons because they are not made of straight line segments. The third shape is not a polygon because it is not closed, while the fourth shape divides the plane into three regions, rather than two.
There are 13 polygons. They are as follows:
Score: (8 • 3) + (1 • 4) + (4 • 5) = 48 points
There are 13 polygons. They are as follows:
Score: (9 • 4) + (4 • 6) = 60 points
There are 13 polygons. They are as follows:
Score: (4 • 3) + (5 • 4) + (2 • 5) + (2 • 6) = 54 points