Introduction
The Sculpture
Courtesy of Martin Fuchs
Marc, Isaac and Isabelle continue to work on their design for the skatepark. Isabelle loves art and thinks that adding some sculpture to the entrance of the skatepark could be cool way to integrate art into the design. Marc and Isaac agree and the three decide to visit a sculpture garden to get ideas. Once they decide on what they want to create, they hope that Mr. Craven, the art teacher, will help them create it with some other classmates.
Upon visiting the sculpture garden, the three notice immediately that there are many different shapes in each sculpture.
Their favorite sculpture is pictured above. Isabelle liked the three dimensional aspect of the sculpture, but did not like that it was all made of triangles.
“Let’s design one with all kinds of polygons,” Marc suggests as they head home.
“That’s a great idea! Which ones should we use?” Isabelle asks.
“What is a polygon anyway?” Isaac interrupts.
Marc and Isabelle look at him. Isaac has not been paying attention in math class.
Before Marc and Isabelle fill in Isaac, what do you know about polygons? Can you define them? Which ones should the trio use in their sculpture? Pay attention in this lesson and you will learn all about polygons.
What You Will Learn
By the end of this lesson, you will be able to demonstrate the following skills.
- Classify polygons.
- Distinguish between regular and irregular polygons.
- Relate sides of polygons to the sum of the interior angles.
- Relate sides of polygons to the number of diagonals form a vertex.
Teaching Time
I. Classify Polygons
This lesson begins talking about polygons in specific detail. In the last two lessons, we worked with triangles and with quadrilaterals. Triangles and quadrilaterals are also polygons; we just haven’t been describing them in this way yet. This lesson will help you to understand how to identify polygons as well as learn some valuable information about them. Polygons are everywhere in the world around us and you will be working with polygons in many ways for a long time.
What is a polygon?
A polygon is a simple closed figure formed by three or more segments. A triangle is a polygon and a quadrilateral is a polygon too. Here are three pictures of polygons.
Polygons
You can see that all three of these figures are simple closed figures that are created by three or more line segments.
Not a Polygon
These figures are not polygons. A polygon does not have a curve in it. The first two figures have curves in them. The third figure is not a closed figure. The last figure has sides that overlap. A polygon does not have sides that overlap.
There are several different types of polygons. Some of them you may have heard of before.
What are some different types of polygons?
1. Triangle – has three sides
2. Quadrilateral – has four sides
3. Pentagon – has five sides
4. Hexagon – has six sides
5. Heptagon – has seven sides
6. Octagon – has eight sides
7. Nonagon – has nine sides
8. Decagon – has ten sides
These polygons can be seen in real life all the time. Look at the following pictures and determine which polygon is pictured.
1.
2.
3.
Take a few minutes to check your work
II. Distinguish Between Regular and Irregular Polygons
Now that you have been introduced to the different types of polygons, it is time to learn about classifying polygons. All polygons can be classified as regular or irregular polygons. You have to understand the difference between a regular or irregular polygon to classify each shape. Let’s learn how we can tell the difference between them.
What is a regular polygon?
A regular polygon is a polygon where all of the side lengths are equal. In other words, the polygon is an equilateral polygon where all the side lengths are congruent.
Let’s look at an example.
Example
This triangle is a regular triangle. All three side lengths are congruent. Here is an example of an irregular polygon.
Example
By counting the sides, you can see that this is a five sided figure. It is a pentagon. However, the sides are not congruent. Therefore, it is an irregular pentagon.
Irregular polygons have side lengths that are not congruent.
Now it’s time for you to practice. Name each figure as a regular or irregular polygon.
1.
2.
Take a few minutes to check your work with a friend.
III. Relate Sides of Polygons to the Number of Diagonals from a Vertex
We can divide polygons into triangles using diagonals. This becomes very helpful when we try to figure out the sum of the interior angles of a polygon other than a triangle or a quadrilateral.
Look at the second piece of information in this box. The sum of the interior angles of a quadrilateral is . Why is this important? You can divide a quadrilateral into two triangles using diagonals. Each triangle is , so the sum of the interior angles of a quadrilateral is .
Let’s look at an example of this.
Example
Here is one diagonal in the quadrilateral. We can only draw one because otherwise the lines would cross. A diagonal is a line segment in a polygon that joins two nonconsecutive vertices. A consecutive vertex is one that is next to another one, so a nonconsecutive vertex is a vertex that is not next to another one.
How do we use this with other polygons?
We can divide up other polygons using diagonals and figure out the sum of the interior angles.
Example
Here is a hexagon that has been divided into triangles by the diagonals. You can see here that there are four triangles formed. If sum of the interior angles of each triangle is equal to , and we have four triangles, then the sum of the interior angles of a hexagon is:
We can follow this same procedure with any other polygon.
What if we don’t have the picture of the polygon? Is there another way to figure out the number of triangles without drawing in all of the diagonals? The next section will show you how using a formula with the number of sides in a polygon can help you in figuring out the sum of the interior angles.
IV. Relate Sides of Polygons to Sum of Interior Angles
To better understand how this works, let’s look at a table that shows us the number of triangles related to the number of sides in a polygon.
Do you see any patterns?
The biggest pattern to notice is that the number of triangles is 2 less than the number of sides. Why is this important? Well, if you know that the sum of the interior angles of one triangle is equal to 180 degrees and if you know that there are three triangles in a polygon, then you can multiply the number of triangles by 180 and that will give you the sum of the interior angles.
Here is the formula.
number of sides
sum of the interior angles
You can take the number of sides and use that as . Then solve for the sum of the interior angles.
Let’s try this out.
Example
What is the sum of the interior angles of a decagon?
A decagon has ten sides. That is our measurement. Now let’s use the formula.
Our answer is that there are in a decagon.
Try a few of these on your own.
- The sum of the interior angles of a pentagon
- The sum of the interior angles of a heptagon
Check your work with a neighbor. Did you use the formula to solve for the sum of the interior angles?
Real Life Example Completed
The Sculpture
Courtesy of Martin Fuchs
Here is the original problem once again. Read the problem and underline any important information.
Marc, Isaac and Isabelle continue to work on their design for the skatepark. Isabelle loves art, and thinks that adding some sculpture to the entrance of the skatepark could be cool way to integrate art into the design. Marc and Isaac agree and the three decide to visit a sculpture garden to get ideas. Once they decide on what they want to create, they hope that Mr. Craven, the art teacher, will help them create it with the help of some other classmates.
Upon visiting the sculpture garden, the three notice immediately that there are many different shapes in each sculpture.
Their favorite sculpture is pictured above. Isabelle liked the three dimensional aspect of the sculpture, but did not like that it was all made of triangles.
“Let’s design one with all kinds of polygons,” Marc suggests as they head home.
“That’s a great idea! Which ones should we use?” Isabelle asks.
“What is a polygon anyway?” Isaac interrupts.
Marc and Isabelle look at him. Isaac has not been paying attention in math class.
Marc, Isabelle and Isaac want to design a sculpture of polygons. A polygon is a closed figure made up of at least three line segments.
Once they fill Isaac in on how to define a polygon, the three students begin to list out different types of polygons.
Triangle
Square
Rectangle
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
After a lot of negotiation, here is a rough sketch of their sculpture design. Can you identify each polygon?
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Polygon
- A simple closed figure formed by three or more line segments.
- Pentagon
- five sided polygon
- Hexagon
- six sided polygon
- Heptagon
- seven sided polygon
- Octagon
- eight sided polygon
- Nonagon
- nine sided polygon
- Decagon
- ten sided polygon
- Regular Polygon
- polygon with all sides congruent
- Irregular Polygon
- a polygon where all of the side lengths are not congruent
- Congruent
- exactly the same or equal
- Diagonal
- a line segment in a polygon that connects nonconsecutive vertices
- Nonconsecutive
- not next to each other
Technology Integration
Khan Academy Sum of Interior Angles of a Polygon
James Sousa, Introduction to Polygons
James Sousa, Classifying Polygons
James Sousa, Interior and Exterior Angles of a Polygon
Other Videos:
- http://www.mathplayground.com/mv_polygon_angle_sum.html – A very clear video on finding the sum of the interior angles of a polygon using diagonals and triangles
Time to Practice
Directions: Determine whether or not each image is a polygon. If yes, write polygon, if no, write not a polygon.
1.
2.
3.
4.
5.
6.
7.
Directions: For numbers 8 – 14, go back and use each figure in 1 – 7. Explain why it is or why it is not a polygon.
8.
9.
10.
11.
12.
13.
14.
Directions: Look at each image and name the type of polygon pictured.
15.
16.
17.
18.
19.
20.
Directions: Name the number of diagonals in each polygon.
21.
22.
23.
Directions: Use the formula to name the sum of the interior angles of each polygon.
24. Hexagon
25. Nonagon
26. Decagon
27. Pentagon