Geometer's Sketchpad Activity
Topic Index | Geometry Index | Regents Exam Prep Center

TASK:  Finding the sum of the interior angles of a polygon.
On this page you will find directions for allowing students to discover the formula for the sum of the interior angles of a polygon using the software package Geometer's Sketchpad (GSP).  The directions shown here are applicable to GSP version 4 or higher.

Student Directions

Part 1:

Finding the sum of the interior angles of a polygon.

 
Geometer's Sketchpad Directions:
(Specific directions on "how to" appear at the bottom of this page.)
  1. Set preferences to allow for the automatic labeling of points and for angle measures in degrees.
  2. Construct quadrilateral ABCD
  3. Choose a vertex of the quadrilateral and draw all diagonals from that point.
  4. Record the number of triangles formed in the table (as seen below).
  5. Using GSP, verify that the sum of the angles in a triangle equals 180 degrees by finding the size of each angle of the triangle and then finding the sum.
  6. Find the total number of degrees of the interior angles of the polygon you are investigating.
  7. Repeat steps 2 - 6 for each additional polygon listed in the table below, or until you can establish a pattern.
Polygon Sides Triangles created by diagonal(s) Sum of the interior angles
triangle

3

1

180 degrees
quadrilateral 4    
pentagon 5    
hexagon 6    
septagon 7    
octagon 8    
n-gon n    

Query:  Examine your table.  Can you determine a rule which can be used to predict  the sum of the interior angles of any polygon?  Write your answer in the row labeled "n" in the table.

Part 2:

Using your prediction (rule), what would be the sum of the interior angles of a decagon?

Verify your answer using the GSP software:

  1. Draw a decagon.  Do not draw the diagonals.
  2. Select and measure each angle of the polygon.
  3. Using the calculator tool, compute the sum of these angles.
  4. Does the sum agree or disagree with your prediction?  State why it does or does not agree.

 

 

Basic "How-To" Directions for Using GSP 4+:
 (Beginning directions for students who are starting to learn to make their own GSP drawings.)
(Directions for use with Geometer's Sketchpad 4.0 or higher.)

1. Set Preferences
  For GSP 4.+ - Choose EDIT, Preferences.  Under the Units tab, set the angle measure to degrees.  Under the Text tab, under "Show Labels Automatically", check the box "For All New Points" to engage the automatic labeling of points. (If you would rather label the points on your own, do not engage automatic labeling.  Simply use the Text tool (the "A") to label points as needed.)
 
2. Construct a Line Segment:
a.  Using the point tool, , hold down the shift key and plot 2 points.  Holding the shift key down will insure that the points remain "selected" for the next step.  Clicking sequentially will also select objects.
b.  Under the CONSTRUCT menu, choose SEGMENT. 
                   

 
3. Construct a Polygon:   
a.  Using the point tool, holding down the shift key and plot points in either a clockwise or counter-clockwise direction.
b.  Select SEGMENT under the CONSTRUCT menu and a polygon will be drawn.

 
4. Construct a Diagonal:
a.  Using the select tool, , select (by clicking) 2 non-consecutive vertices.
b.  Under the CONSTRUCT menu, choose SEGMENT to form a diagonal.

 
5. Use the Calculator:
a.  Under the MEASURE menu select CALCULATE.
b.  Reminder:  The asterisk (*) is used for multiplication.

 
6. Calculate the Measure of an Angle: 
a.  Using the select tool, select three consecutive vertices of the polygon.  (Remember the vertex should be selected as the second point.)
b.  Under the MEASURE menu, select ANGLE.

 
7. Calculate the Sum of the Angles:
a.  Select CALCULATE from the MEASURE menu.
b.  Select each measurement on the sketch.  As a measurement is selected, it will appear in the calculator.  Enter the "plus" between measurements to create the sum. 
c.  Hit OK and the sum will appear in your sketch.