3.4  Polygons 

 

Polygons:

  • A polygon is a closed plane figure formed by three or more
    segments. Each segment intersects exactly two other segments
    only at their endpoints.  No two segments with a common
    endpoint are collinear.
      
        
       
  • A polygon is concave if any diagonal can be drawn between two nonconsecutive vertices and a part of that segment lies outside of the
    polygon.
        
      
        
  • A polygon is convex if all diagonals can be drawn between two nonconsecutive vertices and stay inside of the polygon.
      
      
      
      
  • These are not polygons
    




  
  

Types of Polygons:

Any polygon can be identified as one of these four classifications:
  • A polygon is irregular if it all sides are not congruent and
    all angles are not congruent.
  • A polygon is equilateral if all sides are congruent.
  • A polygon is equiangular if all angles are congruent.
  • A polygon is regular if all sides are congruent and
    all angles are congruent. 

    Note: we do not need to identify a regular polygon as equilateral and equiangular, since it is both by definition.

Names of Polygons:

Polygons are named by the number of sides they have. The following chart shows some common polygons and their names.  If a polygon does not have a specific name, it can be called an n-gon.  For example, if we have a 13 sided polygon we would simply call it a "13-gon".

Polygon Angle Theorems:

  • Formula for the Sum of the Interior Angles of a Convex Polygon:

    To find the sum of the interior angles of a convex polygon, use the formula below, where n is the number of sides of the polygon.

              (n - 2)180º
       
           
      
  • Formula for Interior Angle Measure of a Regular Polygon

    To find the measure of each interior angle of a regular polygon,
    use the formula below, where n is the number of sides of the polygon.

              (n - 2)180°
                     n

        
  • Formula for Exterior Angle Measure of a Regular Polygon

     To find the exterior angle measures of a regular polygon,
    use the formula below, where n is the number of sides in the polygon.

              360°
                n
     

  
  
  

  
  
  
  
  

Example A

Question:

Name and classify each polygon, determine whether it is equiangular, equilateral, regular, irregular, convex or concave.
          


   
Answer:


Watch the video:


Name: Pentagon
equiangular
convex		 Name: Heptagon
equilateral
concave		 Name: Dodecagon
irregular
convex		 Name: Quadrilateral
equiangular
convex

Quick Question...

Instructions:

  • Click on the name & classifications of the given polygon
  • Click on the "Stopwatch" for a 30 second timer
  • Use the slider to select the numbers of questions

To use the app:

  • Clickto begin, click for full screen mode
  • Clickto check your answer.
  • Clickto goto the next problem
  • Clickto stop the app

  

 

Example B

Question:

In the following polygon, what is the sum of the interior angles in the regular hexagon UVWXYZ?

                


Watch the video:
Solution:

    
 

Quick Question...

Instructions:

  • Click on the name &  find the sum of the interior angle measures
  • Click on the "Stopwatch" for a 30 second timer
  • Use the slider to select the numbers of questions

To use the app:

  • Clickto begin, click for full screen mode
  • Clickto check your answer.
  • Clickto goto the next problem
  • Clickto stop the app

  

Example C

Question:

In the following polygon, what is the mZUV in the regular hexagon UVWXYZ?

                


Watch the video:
Solution:

 

Quick Question...

Instructions:

  • Click on the name &  find the interior angle measure
  • Click on the "Stopwatch" for a 30 second timer
  • Use the slider to select the numbers of questions

To use the app:

  • Clickto begin, click for full screen mode
  • Clickto check your answer.
  • Clickto goto the next problem
  • Clickto stop the app

  

Example D

Question:

In the following polygon, what is the measure of the exterior angle at ZUV
in the regular hexagon UVWXYZ?

                


Watch the video:
Solution:

    
 

Quick Question...

Instructions:

  • Click on the name & find the exterior angle measure
  • Click on the "Stopwatch" for a 30 second timer
  • Use the slider to select the numbers of questions

To use the app:

  • Clickto begin, click for full screen mode
  • Clickto check your answer.
  • Clickto goto the next problem
  • Clickto stop the app

  

 

 

Practice Problems

A.  Name and classify each polygon on the right, determine whether it is
      equiangular, equilateral, regular, irregular, convex or concave.
     
B. In the regular polygon ABCDEFGHIJ on the right, find the
     interior angle measure of DEF .
  
  
  
  
C. In the regular polygon ABCDEFGHIJ on the right, find the
     exterior angle measure of HIJ,
  
  
  
  
D. In the regular polygon ABCDEFGHIJ on the right, find the
     sum of the interior angles.

     		 

 

Assignment:

For each given polygon do the following:  
  • Name the polygon.
  • Classify each polygon as either: equiangular, equilateral, regular, irregular.
  • Determine if the polygon is convex or concave.
  • If possible, find the sum of the interior angles.
    If you can not find the measures, explain why you can not.
  • If possible, find the measure of each interior angle.
    If you can not find the measures, explain why you can not.
  • If possible, find the measure of each exterior angle.
    If you can not find the measures, explain why you can not.
  
1.   
   
2.   
3.   
4.   
5.   
  
For each of the following problems use Geogebra to create the polygon with the following:
  
  • Identify congruent sides
  • Identify congruent angles
  • Identify one interior angle measure, for equiangular polygons
  
6.  Create a Regular 14-gon
   		 
7.  Create an equiangular, equilateral, convex quadrilateral
   		 
8.  Create a concave Nonagon
   		 
9.  Create an irregular Hendecagon
   		 
10.  Create an equilateral concave Decagon