Now that we have a better understanding of  regular polygons , let's construct the simplest regular polygon on Geometer's Sketchpad.
 
 

Do you have an idea of what is the simplest regular polygon?

_______________
 

Let's Begin!
 

1.  First, under Display, select Preferences and mark Autoshow labels for points only.  Also, make sure that the precision for distances is Units.

2.  Then, make a circle.  It should have a center marked as A and a point on the circle labeled B.  If needed, move this point B such that it is on top of the circle.

Now, we are going to construct an equilateral triangle in a circle by constructing three isosceles triangles within this triangle.  Since a circle is 360 degrees, what do the central angles of these three triangles have to measure in order to fit inside the circle?

central angle:_______________
 

Hint1

3.  Next, select the circle and go under Construct and select Point On Object. Do this a total of two times.  Then, connect these points to the center of the circle.

4. Highlight each angle and measure all the angles at the center.  If they are not equal to 120 degrees, move the points so that they are all 120 degrees.  Your construction should look like the Figure1.1.
 
 

5.  After you have measured all the angles, connect the chords.  Your figure should look like the Figure1.2.
 
 
 

6. Now, let's calculate the base angles.  Since you know the value of the central angle, what should the value of the base angles be?

base angle:_______________

Hint2
 

7.  With the information about the base angles of the equilateral triangle, can you tell me what is the sum of the interior angles of this triangle?

sum of interior angles in a triangle:_______________
 
 

Part2:  Making a Square

Now, let's increase the number of sides in a regular polygon.

1.  Open a New Sketch and draw a circle with the point B on the circle pointing at the top.  Since a circle is 360 degrees and we need to fit four isosceles triangles into the circle, what should be the measurement of the central angles?

central angle:_______________
 

Hint3
2.  Start making these angles by placing three different points on the circle.  Then, check to make sure that the central angles measure what you calculated in the question above.

3.  Connect the chords to form the bases of these triangles.  Then, record the value of the base angles in the space below.
 

base angle:_______________

4.  With the information about the base angles of the equilateral triangle, can you tell me what is the sum of the interior angles of this square?

sum of interior angles:_______________
 

Now, it's your turn!  Make another regular polygon by increasing the number of sides.  You can continue onto a five-sided regular polygon (pentagon) or any other n-polygon.

Please record your directions on how you constructed your regular polygon below.  Also, find the values of the central angle and base angles.  Do you notice a change in these values as you increase the number of sides?

central angle:_____________

 base angle:______________

sum of interior angles:______
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Now, can find the sum of the interior angles of any regular polygon?  If so, can you provide an equation for the general form of the finding the interior angles of any regular polygon?
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If you are having trouble, please clik on Help.

Help2
Now that you know how to find the interior angles of a regular polygons, do you think that you could figure out what is the sum of the interior angles of a polygon that is not regular?  For example:  "What is the sum of the interior angles of a star?"

Let's find out by going ot Activity3:

STARS, STARS, STARS

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