   SEARCH HOME Math Central Quandaries & Queries  Question from Julie, a parent: How many diagonals can be drawn from one vertex in a 12-sided polygon? Hi Julie,

Lets start with a definition: a diagonal is a line segment which joins two non-adjacent vertices in a polygon.
Also, every polygon has just as many sides as it has vertices (in other words, an 8-sided figure has 8 vertices)

Now let's consider some simpler shapes than a 12-sided figure.

1. In a triangle, all vertices are adjacent and therefore no diagonals can be drawn. Note: a triangle has 3 sides.

2. In a 4-sided figure, once I choose a vertex, two other vertices are adjacent and only one is non-adjacent, leaving me with only one possible diagonal from my chosen vertex.

3. In a pentagon, once I choose a vertex, two other vertices are adjacent, and two others are non-adjacent, leaving me with two possible diagonals from my chosen vertex.

4. In a hexagon, once I choose a vertex, two other vertices are adjacent, and three others are non-adjacent, leaving me with three possible diagonals from my chosen vertex.

I could continue, but perhaps you have noticed that regardless of how many sides the polygon has, once I choose a vertex, there are three vertices to which I can not connect to make a diagonal (the one I chose, and the two adjacent vertices).

Therefore, in general, for any polygon of n sides, there will be (n - 3) diagonals that can be drawn from one vertex.

Hope this helps,
Leeanne     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.