Learning Math: Geometry
Polygons Part A: Hidden Polygons (20 minutes)
Session 3, Part A
In This Part
- Identifying Polygons
- Finding Polygons
Identifying Polygons
Polygons are two-dimensional geometric figures with these characteristics:
- They are made of straight line segments.
- Each segment touches exactly two other segments, one at each of its endpoints.
- They are closed — they divide the plane into two distinct regions, one “inside” and the other “outside” the polygon.
These shapes are polygons:
These shapes are not polygons:
Problem A1
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
Finding Polygons
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:
- 3 points for each triangle
- 4 points for each quadrilateral
- 5 points for each pentagon
- 6 points for each hexagon
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.
Problem A2
How many polygons can you find in the following figure?
Problem A3
How many polygons can you find in the following figure?
Problem A4
How many polygons can you find in the following figure?
Notes
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.
Solutions
Problem A1
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.
Problem A2
There are 13 polygons. They are as follows:
- Four small triangles, each defined by one side of the rectangle and two halves of the diagonals (e.g., XYV)
- Four pentagons, each a complement of one of the small triangles (e.g., VYZWX)
- Four large triangles, each defined by two sides of the rectangle and one of the diagonals (e.g., triangle XZW)
- The rectangle XYZW
Score: (8 • 3) + (1 • 4) + (4 • 5) = 48 points
Problem A3
There are 13 polygons. They are as follows:
- Four small rectangles, all of which share the vertex Q (e.g., SMPQ)
- Four hexagons, each a complement of one of the small rectangles (e.g., PNOLSQ)
- Four larger rectangles, each defined by two small rectangles sharing one side (e.g., MNTS)
- The rectangle MNOL
Score: (9 • 4) + (4 • 6) = 60 points
Problem A4
There are 13 polygons. They are as follows:
- Two small triangles (RUV and TWV)
- Their two complements (hexagons VUSTQR and VWQRST)
- Two quadrilaterals (RQWV and TSUV)
- Their two complements (pentagons VRSTW and VTQRU)
- Two larger triangles (RQT and TSR)
- Three rectangles (RUWQ, USTW, and QRST)
Score: (4 • 3) + (5 • 4) + (2 • 5) + (2 • 6) = 54 points