We have two responses for you

Derek,

The object is called a SECTOR, or more precisely, A SECTOR OF A CIRCLE. (You can have a sector of an ellipse or even of more general shapes). I personally would not use the words vertex and sides in this context, rather I would say that the sector is bounded by two radii and an arc of a circle. The point you refer to as the vertex, I would call the center of the circle. If you prefer to think of the region as a curvilinear triangle, then I would say that the sector consists of three vertices, two straight sides (the radii), and one curved side. I doubt if there is an official terminology, so you should use the words that sound best to you, as long as they are clearly understood by your audience.

Chris

As always with these problem, my first question is:

What is the context? Is there something you are going to do with the counts of vertices and edges?
In mathematics the definitions happen in a context, and are used for certainly kinds of reasoning and predictions.
They also only apply within certain assumptions.

So in 'higher math' one would say that this shape has three vertices and three edges (and one inside, and one infinite outside). This is in the context where 'straightness' does not matter, one is giving the same answer if this is on a rubber sheet and you distort things quite a bit. The branch is called topology. But it also matches how some of the youngest kids process 'shape' - so it could be quite relevant. There are formulae which fit these numbers and help us reason further.

I am trying to think of another context in which these 'answers' would be used.
It is not really the role of math to just give such lists without some reason, and some reasoning.

Walter Whiteley