Download Activity 2: Interior and central angles

Polygons
At a website called PlanetMath, they define a polygon as a “ compact subset of the plane whose
interior is contractible and whose boundary consists of finitely many line segments.” Recall our discussion
concerning the role that definition can play. This is a general math definition. In the first edition of
Discovering Geometry, a high school geometry text, the authors define a polygon as “ a closed
geometric figure in a plane formed by connecting line segments endpoint to endpoint with each segment
intersecting exactly two others. “
Activity 0: What is a polygon?
Question 1: In your group, draw 4 examples of a polygon.
Question 2: In your group, draw 4 examples of a planar shape that is not a polygon.
Question 3: I will be passing a sheet around on which you need to draw your examples in the
appropriate spot. I will then put this sheet on the document projector so that we can all see
everyone’s examples.
Question 4: What are the different ways that we can categorize the polygons? Write these
down as we discuss them. Make sure to include an example of each category.
Question 5: What are the important “parts” of a polygon? List them and label them in a
sketch.
Activity 1: Generating the regular polygons with a compass and protractor.
There is an easy way to generate the regular polygons with a circle.
Step 1: construct a circle with your compass
Step 2: To make a regular polygon with n sides, plot n equally spaced points about the
perimeter of your circle. How are you going to decide what equally spaced means? You need
to be exact.
Step 3: Connect these n points, there is your regular polygon.
Question 1: Using circles, construct regular polygons with n=4, 8 sides. Hint: Can you fold
the circle?
Question 2: With the regular polygons you constructed measure the “central” angles (the
vertex is the circle center and the rays go through a pair of adjacent vertices) with a
protractor or with “common sense”. What is the connection between the number of sides a
regular polygon has, 360 degrees, and the measure of each of central angle? Write an
algebraic relationship.
Question 3: Use the algebraic relationship you discovered in question 2 to fill in the
following chart.
n
3
4
5
6
7
8
9
10
11
12
20
Central
angle
measure
Number
of
central
angles
Question 4: Using your chart in question 3, construct circles with your compass and the
appropriate central angles with your protractors to form regular polygons with n=3, 5, 6
sides.
Question 5: Why would we have to “estimate” with a polygon with n=7 sides?
Activity 2: Interior and central angles
Question 1: In activity 1, you looked at central angles. Find the interior angles of the
polygons you constructed in activity 1. Explain how you did this.
Question 2: What is the algebraic relationship between the interior angle and the “central”
angle? Justify this relationship.
Question 3: Use your work in question 1,2 to fill in the following chart:
n
3
4
5
6
7
8
9
11
Interior
angle
measure
Activity 3: Wrap-up
Question 1: Do all polygons have central angles? Explain.
Question 2: Do all polygons have interior angles? Explain.
Question 3: Draw an example of each “type” of polygon- see the board for the listing of the
different types. Label the parts of each.
Question 4: Describe what a polygon is and is not. Make sure to look back at your work in
activity 0.
Question 5: Define polygon.