My concept of a circle is a continuous, smooth transition from one angle to another, however, on thinking about this further it occurred to me that any portion of a circle, i.e. an arc, can always be simplified in terms of a series of smaller connected straight lines. This kind of makes sense to me because, after all, the most direct route from one point to another is a straight line so a circle is merely a question of resolution, in other words, what appears to us as a curve is in reality a series of straight lines but because we cannot perceive them presents as a curve.
Congratulations, you just invented calculus.
There are a couple of semesters worth of deeper questions in there.
What you you mean by exist?
What do you mean by “direct route” ?
How many straight lines are you thinking of?
How short are these lines?
What do you mean by “straight”?
What do you mean by “continuous”?
What do you mean by “smooth”?
What do you mean by “smaller”?
What do you mean by “connected”?
What do you mean by “resolution”?
This isn’t to be snarky, every one of those questions has puzzled mathematicians, and not all have a resolution. How you answer those questions can lead to a remarkable extension of your universe.
Or drafting.
I did? Thank you. 
Hmmm…my goodness, I seemed to have opened up a can of worms here. 
Well, it seems to me people must have resolved such questions otherwise mathematics would never have gotten off the ground.
Yeah, that’s what I’m talking about - good video.
Yeah, the subject comes up in calculus, but in a way that rather contradicts the assumptions of the OP. A curve (at least as described mathematically) isn’t a series of tiny straight lines; it really is constantly curving. You could pick two points on a curve (far apart or very, very close together) and draw a straight line between them. But calculus can define a line that touches a curve only at a single point. In pure math terms, a circle is always curving.
Now, if the OP wants to make the case that such a curve can’t be constructed in the real world, he’s probably on to something. On a computer screen, your ability to draw a circle is limited by the pixels that make up the display. If you use pencil and paper, the best you could do is to deposit a series of discrete carbon atoms. Finer than that you run into problems with Planck length, beyond which you really can’t tell two points apart. (The pixel size of the universe, in a manner of speaking.)
But is it really a cylindrical can?
Well, okay, but then I’m thinking no matter how many times you can draw a curve between two points, you can then keep sub-dividing such a curve into further straight lines, so the question (I guess) is: which came first the curve or the straight line? To my way of thinking, as a straight line is a simpler and more fundamental geometrical concept than a curve (Occam’s razor?), it would seem to be the straight line, but I could be wrong!
Yes, I agree but that’s a slighly different point, although well made.
LOL…Very good! 
Most are resolved, what I do is encourage you to think about each one a bit. See if you can explain to yourself what you mean by each one, and to consider how much prior definition you need to make your explanation meaningful. Especially take care that you are not going around in a loop, basing one definition on another and the first on the second.
Many of those words have very strict mathematical meanings, but many have multiple definitions, which one to use depending upon the circumstances it is used in.
“Smooth” for instance is going to take you down a multiple set of rat holes. But even “straight” far from trivial.
In general you might want to think of a circle as a curved line. IE take the number line, cut a length of it out, and then curve it around. Then ask yourself how many numbers there are along that segment of line. Continue from there…
You might also like to consider a simple definition for a unit circle: The solutions to the equation x[sup]2[/sup] + y[sup]2[/sup] - 1 = 0
You now have coordinates for all the points (the <x,y> pairs that solve the above.) Consider how many there are of these, and again continue to think about the question…
A circle is just a triangle, for very large values of 3
I was going to suggest there might be a perfect Platonic circle, but then that raises the question of why a sphere isn’t a Platonic solid and that way lies madness and/or more math.
A circle is a figure made up of infinitely many line segments of one point length. If you want to think of it that way. Of course, you can’t really have a segment if you don’t have two points (and all the points in between), but we can consider it the reduction of the concept to the point of merging the two endpoints into one point.
However, it is wrong to think of the circle as being made up of actual segments that are too small to see. If I magnify the circle, I will always see the same thing: a curve, not a straight segment. And I can NEVER magnify the circle to the point that I see the individual points of the circle (they are infinitely small, having no dimension), but no matter how magnified the circle is, it will always be a curve, and not any straight segments. This is because all the points of the circle are equidistant from the center, and if the circle had any segmented parts, some of those points would be closer to the center than others (the hypotenuse of a right triangle being longer than the legs).
[sigh] … I thought I’d scoop everyone with Euclid’s Third Postulate:
I.E. - Given any line segment, a circle can be drawn using the segment as the radius with one endpoint as the center … however this only defines a circle, rather than defending its existence …
The answer to every one of abashed math questions is that sets of infinite points don’t work in the same way as “common sense” tells us. Special rules, procedures, definitions, and postulates have to be created and rigorously adhered to.
As long as you try to apply the thinking of finite points you will fail, as centuries of mathematicians did until Cantor’s set theory tamed infinities. They make perfect sense within that structure. They do not make sense by starting with finites - polygons are sets of straight lines - and trying to make the jump to infinities - the set of points at a certain distance from a center - using the same logic.
Given that pi is an irrational number, I’m pretty sure that means a true circle is impossible to make in the real world. Even if you’re dealing with the smallest particle in existence, you couldn’t arrange them in a complete circle without using a fraction of that particle to close the circle.
Okay, I’ll take a stab.
What you you mean by exist?
That which can be measured.
What do you mean by “direct route” ?
The least measured distance between two points.
How many straight lines are you thinking of?
Infinitely many (mathematically speaking).
How short are these lines?
There is no end to their shortness (mathematically speaking).
What do you mean by “straight”?
That the angle at which a line starts does not alter over the course of its distance between two points.
What do you mean by “continuous”?
The same definition as above.
What do you mean by “smooth”?
No change in direction (so no change in angle).
What do you mean by “smaller”?
Measurably smaller in terms of the number system.
What do you mean by “connected”?
That there exists no gaps between geometric ideas, such as points, lines, curves, etc.
What do you mean by “resolution”?
How small we are able to measure something.
I’m struggling with this, to be honest. I would have thought you could say any curve could be reduced to an infinite numbers of points?