Why is it safe to approximate $2\pi r$ with regular polygons? Considering this question: Is value of $\pi = 4$?
I can intuitively see that when the number of sides of a regular polygon inscribed in a circle increases, its perimeter gets closer to the perimeter of the circle. This is the way Archimedes approximated $\pi$.
However, the slope of the tangent at almost every every point of the circle differs from the slope of the tangent to the polygon. This was basically why $\pi \neq 4$. Actually, only finite number of slopes coincide. 
Then how do we make sure that the limit is $\pi$?
Thanks.

The perimeter of a regular $n$-gon inscribed in a circle of radius $1$ is $2n \sin (\frac{\pi}{n})$, and $ (\frac{n}{ \pi}) \sin(\frac{\pi}{n}) \to 1$ as $n \to \infty.$
