Question 979233: The following polygons are given. All of the polygons are
regular polygons.
Polygon a. Convex 15-gon
Polygon b. Convex 16-gon
Polygon c. Convex 17-gon
Polygon d. Convex 18-gon
Polygon e. Convex 19-gon
Polygon f. Convex 43-gon
Polygon g. Convex 44-gon
Polygon h. Convex 45-gon
Polygon i. Convex 46-gon
Polygon j. Convex 47-gon
1. Which polygon(s) has (have) interior angles that are whole
numbers (a number that is not a fraction or a decimal)? Explain
why it is that way.
2. What happens to the value of the interior angles as the
number of sides of the polygon increases? Explain your answer.
3. What happens to the value of the exterior angles as the
number of sides of the polygon increases? Explain your answer.
4. Explain what happens to the total sum of interior angles
as the number of sides in the polygon changes?
5. Explain what happens to the total sum of exterior angles
as the number of sides in the polygon changes?
Found 2 solutions by Edwin McCravy, solver91311:
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website! The following polygons are given. All of the polygons are
regular polygons.
Polygon a. Convex 15-gon
Polygon b. Convex 16-gon
Polygon c. Convex 17-gon
Polygon d. Convex 18-gon
Polygon e. Convex 19-gon
Polygon f. Convex 43-gon
Polygon g. Convex 44-gon
Polygon h. Convex 45-gon
Polygon i. Convex 46-gon
Polygon j. Convex 47-gon
1. Which polygon(s) has (have) interior angles that are whole
numbers (a number that is not a fraction or a decimal)? Explain
why it is that way.
The sum of the interior angles of a polygon of n-sides is
Since the polygons are regular, all the interior angles are the same,
so each one is that expression divided by n
That must be equal to a whole number, say, W. Since n does not divide
evenly into n-2, it must divide evenly into 180°. So we go through
the list to see which numbers divide evenly into 180°:
Polygon a. Convex 15-gon, yes, since 15 divides evenly into 180°.
Polygon b. Convex 16-gon, no
Polygon c. Convex 17-gon, no
Polygon d. Convex 18-gon, yes, since 18 divides evenly into 180°.
Polygon e. Convex 19-gon, no
Polygon f. Convex 43-gon, no
Polygon g. Convex 44-gon, no
Polygon h. Convex 45-gon, yes, since 45 divides evenly into 180°.
Polygon i. Convex 46-gon, no
Polygon j. Convex 47-gon, no
2. What happens to the value of the interior angles as the
number of sides of the polygon increases? Explain your answer.
    
So the value of the interior angles approaches 180° as the number
of sides of the polygon increases.
3. What happens to the value of the exterior angles as the
number of sides of the polygon increases? Explain your answer.
The sum of the exterior angles of any polygon is 360°. So
each one of a regular polygon is  
So the value of the exterior angles approaches 0° as the number
of sides of the polygon increases.
4. Explain what happens to the total sum of interior angles
as the number of sides in the polygon changes?
 
5. Explain what happens to the total sum of exterior angles
as the number of sides in the polygon changes?
 
Edwin
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
If  is the number of sides and 180) , then the measure of each interior angle is an integer.
As increases, the measure of the interior angles increases because 180}{n}) gets closer to 180 as gets larger. A circle is the limiting shape as increases without bound so a circle is an infinite-sided polygon with an infinite number of vertices with 180 degree interior angles. It was this idea that allowed Archimedes to approximate  by sandwiching a circle between two 96-sided polygons, one inscribed and the other circumscribed.
As increases, the measure of the exterior angles decreases because  gets smaller as gets larger.
As increases, the total of the measures of the interior angles increases because 180) gets larger as gets larger.
As increases, the total of the measures of the exterior angles remains constant because  does not change as gets larger. Another way to put it is, no matter how many increasingly smaller turns you make, you still only go around one circle when you get back to where you started.
John
My calculator said it, I believe it, that settles it
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