What if you were told how many sides a polygon has? How would you describe the polygon based on that information? After completing this Concept, you'll be able to classify a polygon according to its shape and the number of sides it has.
Watch This
James Sousa: Introduction to Polygons
Then watch the first part of this video.
James Sousa: Classifying Polygons
Guidance
A polygon is any closed, 2-dimensional 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 .
Polygons can be either convex or concave . The term concave refers to a cave, or the polygon is “caving in”. All stars are concave polygons.
A convex polygon does not cave in. Convex polygons look like:
A diagonal is a non-side line segment that connects two vertices of a convex polygon.
The red line segments are all diagonals. This pentagon has 5 diagonals.
Whether a polygon is convex or concave, it is always named by the number of sides. Explore the relationship between the number of sides of a convex polygon and its diagonals. Can you complete the table?
Polygon Name | Number of Sides | Number of Diagonals | Convex Example |
---|---|---|---|
Triangle | 3 | 0 | |
Quadrilateral | 4 | 2 | |
Pentagon | 5 | 5 | |
Hexagon | 6 | 9 | |
Heptagon | 7 | ? | |
Octagon | 8 | ? | |
Nonagon | 9 | ? | |
Decagon | 10 | ? | |
Undecagon or hendecagon | 11 | ? | |
Dodecagon | 12 | ? | |
n-gon | (where ) | ? |
Example A
Which of the figures below is a polygon?
The easiest way to identify the polygon is to identify which shapes are not polygons. and each have at least one curved side, so they are not be polygons. has all straight sides, but one of the vertices is not at the endpoint, so it is not a polygon. is the only polygon.
Example B
Determine if the shapes below are convex or concave.
To see if a polygon is concave, look at the polygons and see if any angle “caves in” to the interior of the polygon. The first polygon does not do this, so it is convex. The other two do, so they are concave.
Example C
Name the three polygons below by their number of sides and if it is convex or concave.
The pink polygon is a concave hexagon (6 sides).
The green polygon convex pentagon (5 sides).
The yellow polygon is a convex decagon (10 sides).
Guided Practice
1. Which of the figures below is not a polygon?
2. How many diagonals does a 7-sided polygon have?
3. True or false: A quadrilateral is always a square.
Answers:
1. is a three-dimensional shape, so it does not lie within one plane, so it is not a polygon.
2. Draw a 7-sided polygon, also called a heptagon.
Drawing in all the diagonals and counting them, we see there are 14.
3. False. Only quadrilaterals with four congruent sides and four right angles will be squares. There are many quadrilaterals (such as rectangles, kites, parallelograms, trapezoids, etc.) that are not necessarily squares.
Practice
In problems 1-6, name each polygon in as much detail as possible.
- Explain why the following figures are NOT polygons:
- How many diagonals can you draw from one vertex of a pentagon? Draw a sketch of your answer.
- How many diagonals can you draw from one vertex of an octagon? Draw a sketch of your answer.
- How many diagonals can you draw from one vertex of a dodecagon?
- Determine the number of total diagonals for an octagon, nonagon, decagon, undecagon, and dodecagon.
For 12-14, determine if the statement is true or false.
- A polygon must be enclosed.
- A star is a convex polygon.
- A 5-point star is a decagon