# Is this a polygon

Suppose we take a rectangle and cut out a smaller quadrilateral within it, as shown below:

Would the resulting figure be a polygon? If it is, would it be a concave octagon?

According to what I'm reading in this geometry textbook, a polygon is a plane figure that is formed by three or more segments called sides such that

1. Each side intersects exactly two other sides, once at each endpoint.
2. No two sides with a common endpoint are collinear.

This wasn't a question I found in the textbook or anything. It was just a thought that popped into my head. I find it confusing because it is a very unusual example.

My assumption is that yes, it is a polygon since the figure fits the description. I know that if it is a polygon then it must be concave, since a line containing a segment of the inner quadrilateral would go through the interior of the figure.

I will be thankful for help!

• By this definition, yes this is a polygon. Normally one requires the sides to be a piecewise curve where each piece is a straight line segment. The above doesn't fit the latter description. – Mathematician 42 Feb 18 '18 at 8:17

## 1 Answer

We come up with definitions to describe the things we want to work with, not the other way around. It's probably important to make a decision one way or the other about whether the thing you've drawn is a polygon, but the correct way to make that decision is not "what does the wording in the textbook imply?" but "what would be useful?"

(I suspect that the authors of the textbook did not have such a figure in mind when writing the definition of a polygon, though their definition does seem carefully worded to avoid self-intersecting polygons.)

In this case, the figure you've drawn doesn't have many of the nice properties that other polygons (even concave ones) do. For example, it appears to be an octagon, but the angles of an octagon should add up to $1080^\circ$, and the angles of this figure add up to $1440^\circ$.

We could allow this to be a polygon, call the more-reasonable objects we're used to studying simple polygons, and change our theorems to say things like "The sum of angles in a simple polygon with $n$ sides is $(n-2)180^\circ$." Or we could say that this is not a polygon, but it is a "generalized polygon" or something like that.

(You should really read through the Wikipedia article on simple polygons I linked to above; it's an example of a very carefully thought-out definition.)

Either way, the point is that this has some, but not all, of the properties we're used to seeing in a polygon. What makes things worse is that people often work with polygons on a somewhat intuitive level, so can get hard to tell whether a particular theorem about polygons assumes that their boundary is one continuous curve or not. So you should feel free to call it a polygon - but only if you think things through before assuming that it has a property someone tells you all polygons have.