It is not correct to say that a circle is "an infinite sided polygon" – or more precisely, it would take more clarification about what you mean by "polygon" to even make it mean anything to say that. The definition of "polygon" from Wikipedia:
In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit.
Therefore your reasoning based on this statement is already flawed.
Putting that aside, you'd also have to put a lot more work into making precise what it means for the radius of the circle to be "made infinite". Infinity is not a single concept and most versions of it cannot be treated as a number, so you can't just say $r=\infty$ without a lot of clarification.
Mark's answer is correct, that in complex analysis (and other parts of math) it is sometimes useful to treat circles and lines similarly, but I would say that it is not useful in the context of this discussion.
Putting that aside, you'd have to explain what it would mean for the sides of a polygon to be "infinitely long", and actually prove that "making the circle radius infinite" also makes the "sides" "become infinitely long".
That's just the first sentence, but I hope you see that the general problem is that you need to be very clear and rigorous about the mathematical words you use, and particularly infinity is a tricky concept and prone to causing mistaken reasoning without proper understanding and precision.