How many sides can a polygon have before it is "considered" a circle?

Question

Good day, my family had a dinner discussion about polygons and how many sides a polygon has in relation to the angle measurement you'll get when you measure an "arc" encompassing a "side" of the polygon. Of course this is assuming that all sides are equal.

In this case, we'll get 120 degrees for a triangle, 90 degrees for a square, and the degree measurement decreases as the number of sides increase.

Now, my question is, with this above in mind, up to how many sides can a polygon have? We can have a 360-sided polygon with each side having an arc of 1 degree, but you can go smaller than 1 degree and argue that you can go with 0.000001 degree and have the corresponding number of sides. However, since there are an infinite number of fractions between two numbers, we can go infinitely many times and get n amount of sides based on a very small angle m.

If we go as such, up to how many sides can we get before the shape we have can be considered as a circle?

Answer 1

A polygon is never a circle, no matter how many sides it has.

Answer 2

You and your family are effectively exploring the notion of limits from calculus in a geometric context. Think of this problem in terms of real numbers for a second. If we start at 1 and add $\frac{1}{2}$, and then add $\frac{1}{4}$, and then add $\frac{1}{8}$, etc., how many times would we need to add these terms before the sum was considered to be the number 2? The answer is that for any finite amount of these terms that we add together, our sum will always be less than two. Even so, we can observe that no matter what small number we choose, there will be one of these sums will be less than that small number away from 2.

Now apply this idea to regular polygons and circles.