
[Question]
The sum of the vertex angles of a certain convex polygon is three
times the sum of its central angles. How many sides does the polygon
have?
[Difficulty]
I'm not sure how to approach the problem. Do I draw the polygon?
[Thoughts]
 Glavy


Hi Glavy,
You can't really draw the polygon, because you don't know how many
sides it has. :^D
However, you could draw _some_ polygons, and see if it's true for
those polygons. For example, consider an equilateral triangle. The
sum of the vertex angles is 180 degrees. The sum of the central
angles is 360 degrees.
sides sum of vertex angles sum of central angles
  
3 180 360
Okay, what about a square? The sum of the vertex angles is 4*90, or
360 degrees. And the sum of the central angles is... 360 degrees.
sides sum of vertex angles sum of central angles
  
3 180 360
4 360 360
If you keep going, you should either find out what the answer is, or
that there is no answer.
Does this help?
 Doctor Ian, The Math Forum


Dear Dr. Math,
Wow...thanks to you I think I've managed to answer the question. Ok,
this is what I got. You proved to me that the measure of all central
angles is 360. Also, you showed me that if you multiply the number
of sides to 90, you'll get the sum of the vertex.So therefore what I
did was this:
sum of vertex angles = 3(360)
= 1080
12 times 90 = 1080
Which comes to show that the polygon has 12 sides. Thank you so much!
You're excellent!
 Glavy


Hi Glavy,
>Wow...thanks to you I think I've managed to answer the question.
Well, you got _an_ answer, but it's not the correct one.
>Ok,
>this is what I got. You proved to me that the measure of all central
>angles is 360.
So far, so good.
>Also, you showed me that if you multiply the number
>of sides to 90, you'll get the sum of the vertex.
It's not that simple. Draw a triangle, a rectangle, and a pentagon.
Now pick one vertex on each, and draw all the diagonals that you can
from that vertex.
In the triangle, you'll get no diagonals. And the sum of the vertex
angles is 180 degrees.
In the rectangle, you'll get one diagonal, which divides the rectangle
into two triangles. The sum of the vertex angles will be twice the
sum for a single triangle, or 360 degrees.
In the pentagon, you'll get two diagonals, which gives you three
triangles, for a total of 3*180 = 540 degrees.
So for a polygon with N sides, you'll get a total of (N2)*180 degrees
for the sum of the vertex angles.
>So therefore what I
>did was this:
>sum of vertex angles= 3(360)
> = 1080
>12 times 90 = 1080
>Which comes to show that the polygon has 12 sides. Thank you so much!
>You're excellent!
Give it another try, and let me know what you get.
 Doctor Ian, The Math Forum


Dear Dr. Math,
All central angles = 360
A polygon with N sides would be:
(n2) time 180
180N  360 = 1080
180n = 140
n = 8
therefore, the polygon has 8 sides.
Thanks.
 Glavy


Hi Glavy,
Nicely done! Thanks for letting me know that you got it okay.
 Doctor Ian, The Math Forum


