Finding the Sum of the Exterior and Interior Angles of a Polygon

David Drymiller                Morgan Park High School
                               1744 W. Pryor Ave.
                               Chicago IL 60643
                               (312) 535-2550

Objectives:

     To find the sum of the interior angles and the sum of the exterior angles
       of any polygon.
     To review linear measurement to the nearest sixteenth of an inch and 
       angle measurement to the nearest degree. 
     To construct a polygon and its exterior angles given the number of sides.

Materials needed:

     For each group of four students the following is needed:
       a yardstick or meterstick, 
       a large demonstration type protractor,
       chalk and instruction data record sheet.
     
     For each student the following is needed:
       ruler, protractor.

Strategy:
                
     Each group draws a large 7, 8, 9, 10, 11 or 12 sided polygon on the floor 
or sidewalk with unequal sides.  Measure each interior angle of the polygon to 
the nearest degree and record the results.  Measure the length of each side and 
record the results.  Find the sum of the interior angles of the polygon and 
record the answer.  Extend each side of the polygon forming an exterior angle at 
each vertex.  Measure each angle to the nearest degree and record the result.  
Find the sum of the exterior angles of the polygon and record the result. 
     As a class, review the fact that the sum of the interior angles of a 
triangle is 180.  Divide 4, 5, 6, 7, 8, 9, 10, 11, and 12 sided polygons into 
triangles showing the sum of the interior angles is 2x180, 3x180,..., 10x180, 
respectively.  Generalize to the sum of the interior angles of a n sided polygon 
is (n-2)180.  Draw a 7 sided polygon and its exterior angles and label the 
angles.  Mark one end of the meterstick and place its center on the first 
vertex, slide the center along the side of the polygon to the next vertex, 
continue until arriving back at the starting vertex.  The path of the marked end 
of the meter stick is a circle.  Generalize to the sum of the exterior angles of 
a polygon is 360.  Each student draws a 3, 4, 5, 6, 7, or 8 sided polygon and 
its exterior angles to the edge of the paper.  Mark an arc on each exterior 
angle and cut each out.  Place all of the vertices of the exteriors angles 
together forming a circle.  Generalize to the sum of the exterior angles of a 
polygon is 360.  
     Follow the above activity with an octagonal work sheet having the students 
record the measure of each angle and side on the figure.  Place the sum of the 
interior angles in the center of the octagon.  Tape the octagon onto  a second 
sheet.  Draw the exterior angles of the octagon on the second sheet and record 
each measure on the angle.  Place the sum of the exterior angles on the bottom 
of the second sheet.