Oh, dude, a circle has zero vertices. I mean, it's just a smooth, round shape with no corners or points sticking out. So, if you were hoping to count some vertices on a circle, you might be out of luck. But hey, at least you can enjoy its perfect symmetry, right?
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Zero vertices
None but it's possible to construct shapes within a circle that have vertices.
The shape that has no vertices and is round is a circle. The circle is the only shape that does not have vertices.
Depending on how you want to define vertices, a circle either has no vertices or the circle is composed of infinite vertices. A vertex is where two straight edges meet. Since a circle is continuously curved you can easily argue that it has no straight edges to meet and form a vertex. From a limits standpoint though it has infinitely many straight segments - but they are are all of length 1/∞
In this context, an ellipse is equivalent to a circle: it has one edge and one face and no vertices.