POLYGONS MADE TO ORDER

Malone, Loretta                    Washington High School
                                   1-312-646-4860
                           

OBJECTIVES

1. To construct a regular polygon inscribed in a circle by using isosceles triangles 
   with vertex angles at the center of the circle and legs as radii. 

2. To determine the values of the vertex angles (central angles), the values of the 
   base angles, the values of the inscribed angles, and the sum of the angles of any 
   regular polygon. 

3. To allow the students to discover that as the number of sides of a polygon 
   increases, the polygon appears to look like a circle. 

EQUIPMENT AND MATERIAL

1. Compass              
2. Protractor
3. Straight edge
4. Worksheets
5. Overhead Projector and projector pens
6. Colored transparencies with regular polygons

RECOMMENDED STRATEGY

A. Review the concepts of triangles and circles. 
   1. Isosceles triangle 
   2. Base angles and vertex angle 
   3. Circle 
   4. Radius 
   5. Chord 

B. Draw the circle. 

C. Construct an equilateral triangle inscribed in a circle. 
     Find the value of a vertex angle by dividing 360 by 3. Draw a radius and 
construct the vertex angle using the value. Construct the remaining two congruent 
angles using radii as sides. Connect the chords. (three isosceles triangles are 
constructed.) Place the value of the central angle on worksheet 1. 

D. Calculate the values of the base angles. 
     Find the values of the base angle by subtracting the value of the measure of the 
central angle from 180, then divide that value by 2. Place the value on worksheet 2. 

E. Increase the number of sides in a regular polygon. 
     Construct a square inscribed in a circle. Use four radii by constructing four 
congruent central angles. (360/4) Draw the chords (bases) formed by the isosceles 
triangles. Place the value of the new central angle on worksheet 1. Place the value 
of the new base angle on worksheet 2. 

F. Continue to increase the number of sides (5,6,7,8,...12,...) in the regular 
   polygon. 
     Find the value of each new central angle and notice the decrease. Find the value 
of the new base angles and notice the increase. 

G. Find the sum of the base angles of any regular polygon. 
     Use the number of isosceles triangles constructed inside the circle. Find the 
sum of the base angles.(worksheet 2) Multiply that number by the number of sides of 
the polygon. Place the new value in worksheet 3. 

     Students can easily learn to understand what a regular polygon is. They will 
begin to notice the values of the angles; i.e., the decrease in the central angles 
and the increase of the inscribed angles.  Students will start to question what is 
the largest regular polygon. 

Worksheet 1: The values of the central angles for each different polygon 
Worksheet 2: The values of the base angles in each set of isosceles triangles 
Worksheet 3: The sum of the measures of the angles in any regular polygon