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### Counting Sides by Counting Diagonals

```Date: 06/04/2002 at 07:41:53
From: Sunita
Subject: Sides in a Polygon

I wish to have a formula that can tell me how to calculate the number
of sides of a polygon if I know the number of diagonals in that
polygon. For example, I know that the number of diagonals in a
polygon is 44. How many sides does this polygon have?
```

```
Date: 06/04/2002 at 07:48:45
From: Doctor Ian
Subject: Re: Sides in a Polygon

Hi Sunita,

Suppose you have a polygon with N vertices.  From each vertex,
you can draw a diagonal to (N-3) other vertices, so the number of
directed diagonals is N(N-3).

However, when you do this, you end up with diagonals going in
both directions, so you want to cut the number in half.  So the
number of undirected diagonals is N(N-3)/2.  Convince yourself
that this works for any N.

To go from the number of diagonals to the number of vertices, you
just go in the other direction:

44 = N(N-3)/2

88 = N(N-3)

11 = N

So the polygon has 11 sides.

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Triangles and Other Polygons
Middle School Triangles and Other Polygons

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