What is an N-gon?

Date: 06/01/98 at 18:17:23
From: Missy
Subject: I'm confused!

Dear Dr. Math,

I can't figure out the answer to this question. 

Consider the following pattern: In an N-gon, n-3 diagonals can be 
drawn from one vertex. When N = 4, to what kind of polygon does the 
pattern refer?

I don't get the part of "n-3 diagonals." Does that mean that to find 
the answer, I have to subract 4 from 3?

Missy

Date: 06/01/98 at 19:00:04
From: Doctor Barrus
Subject: Re: I'm confused!

Hi, Missy!

I'm going to split the statement up and try to explain each part.

First: "In an N-gon, n-3 diagonals can be drawn from one vertex."

I think that "N" and "n" mean the same number. What this means, then, 
is this: Say you have a polygon with n sides (this is called an 
n-gon). Now pick any vertex (corner). We're going to count how many 
diagonals (lines connecting two vertices that don't touch the sides of 
the n-gon) we can draw from that vertex. For example, let's look at a 
hexagon:

       1_______2
       /       \     Now pick one of the corners (Let's suppose I 
      /         \    choose corner number 1). Draw lines connecting
     /           \   that corner with the other corners that aren't
    6\           /3  already connected to corner 1. These corners are
      \         /    3, 4, and 5. The lines that you've drawn are the
       \_______/     diagonals drawn from vertex 1.
       5       4

What the statement is saying is that from any vertex, or corner, of an 
n-gon, you can draw n-3 diagonals (in our example, we could draw 
n-3 = 6-3 = 3 diagonals to corners 3, 4, and 5). To see why this is, 
look at this: 

 - We have n points as the vertices.
 - We can't draw a line connecting a point with itself.
 - A point is already connected to two other points by the sides of 
   the n-gon.

So we have n - 1 - 2 = n - 3 points to connect to. Therefore, there 
are n - 3 diagonals.

Second: "When N = 4, to what kind of polygon does the pattern refer?"

This is just asking what kind of polygon has 4 sides. You'll also want 
to look at how many diagonals the formula says you can draw from one 
vertex, and check that by actually drawing the diagonals.

Well, I hope this helps. Good luck!

-Doctor Barrus,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/