Proving a circle’s sides approach infinity, is my proof correct?

Question

Before, I tried to prove that when an equalateral polygon’s sides increases, the ratio of the circumference/diameter (or perimeter/height) gets closer to $\pi$. So I made an equation to divide the perimeter of a polygon by its height, which is x/tan((((x-2)*180)/x)/2)=pi (I simplified it), and $x$ represents how many sides the polygon has (except for a triangle) and I was correct, when $x$ increases the answer becomes closer and closer to $\pi$, so when a polygon with $x$ number of sides approaches infinity the circumference/diameter approaches $\pi$. So in the equation x/tan((((x-2)*180)/x)/2)=ou isn’t the value of $x$ how many sides a circle has? I don’t think it’s possible to simplify but if it were possible, then maybe there could be an expression written to show how many sides a circle has.

Answer 1

Your "simplified" expression actually simplifies further: \begin{align} \frac{x}{\tan\left({\left(\dfrac{(x-2)\times180^\circ} {x}\right)}/2\right)} &= \frac{x}{\tan\left({\dfrac{(x-2)\times180^\circ}{2x}}\right)} \\ &= \frac{x}{\tan\left({\dfrac{180x^\circ - 360^\circ}{2x}}\right)} \\ &= \frac{x}{\tan\left(90^\circ - \dfrac{180^\circ}{x}\right)} \\ &= \frac{x}{\cot\left(\dfrac{180^\circ}{x}\right)} \\ &= x \tan\left(\dfrac{180^\circ}{x}\right). \end{align}

So you are calculating the perimeter of a polygon circumscribed about a circle of diameter $1$.

It is true that as $x\to\infty$, the perimeter approaches $\pi$ and the shape of the polygon approaches a circle, so you could say that in as the shape approaches the circle the number of sides approaches infinity. However, the existence of a limit tells us nothing about what happens at the limiting condition, only what happens near the limiting condition.

The circumference of the circle is the length of a curved path, for which we need a definition of how to measure that length. It happens that the commonly used definition for length of a curved path says that the limit of the polygons' perimeter is also the circumference of the circle. But that definition says nothing about the "sides" of a curved path or in particular the "sides" of a circle.

By the usual definitions, a circle has no sides at all (in the sense of "side" that we use for polygons), because a side is a straight path between two distinct points, and there is no part of the circle that matches this description.