Circle having sides?

[Eddie]

Now, I know people often debate the idea of a circle having sides. Does a circle have zero sides? Does a circle have one side? Does it have two sides (inside and outside)? Does it have infinitely many sides? Etc.

Most people claim that the definition of a side is that it must be straight and therefore a circle does not have sides, because a circle is always perfectly round.

Now, we know in the Cartesian system, the equation of a circle centered at the origin is x^2 + y^2 = r^2.

Through simplification, we can see that y = ±(r^2 - x^2)^(1/2).

Now, we know that the formula for the length of a smooth curve [L = S(1 + (dy/dx)^2)^(1/2)dx where the limits of integration are the starting and ending point of the curve you want the length of] is crafted from the distance formula and works because of the local linearity of functions. It is taking an infinite sum of infinitely small distances.

Without loss of generality, we will let the circle we are dealing with be a unit circle centered at the origin.

y = ±(1 - x^2)^(1/2).

Now, we will split it up into two parts, one will choose the positive side, the other will choose the negative side.

So, dy/dx = (-x)(1 - x^2)^(-1/2) and dy/dx = (x)(1 - x^2)^(-1/2)

Now, we plug each one into the length formula and solve, and we get 2pi, which is not surprising because that is what we know the circumference of a unit circle to be.

Since the length of a smooth curve formula can be used (and its results are accurate) for a semi-circle, wouldn't that imply that the semi-circle (and therefore the circle) is locally linear if you "zoom in" small enough?

If that is the case, wouldn't it therefore be a good assumption to make that a circle has infinitely many sides?