I have read about how they calculate the value of $\pi$ using polygons.
A hexagon inside a circle gives an approximation of $\mathbf{\pi} = 3.00$, then doubling the number of the number of polygon sides gives a closer approximation of $3.10582$, and if we keep doubling the amount of sides the value for $\pi$ starts to plateau. At $1536$ sides, we get the first six digits of $\pi = 3.14159$.
Would it be fair to say that the value of $\pi$ that we use is just a very close approximation? To me it seems that no matter how many sides a polygon has, its perimeter will always be less than the circumference of a circle that encloses it.
Are there any other proofs for $\pi$ that are 'not too technical'?