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.

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Didn't Archimedes do it with area, not perimeter? –  Steven Taschuk Jun 13 '14 at 15:59
All my internet search says perimeter. –  ThePortakal Jun 13 '14 at 16:04
Hm. My mistake, then. As for your question, I think one important issue will be how you define the length of a curve; nowadays it'd usually be defined exactly by this kind of approximation-by-polygons process, so there's kind of nothing to show. –  Steven Taschuk Jun 13 '14 at 16:11
Both area and perimeter lead to the same approximation. Just cut it into a bunch of triangles, each traingle's area is $0.5 {\tt Base} * {\tt Height}$, $Height \approx Radius$, $\sum (Bases) = Perimeter$... –  PA6OTA Jun 13 '14 at 16:13

The reason why we know the Archimedean approximation works when we also know the 'troll' (rectilinear) approximation doesn't is because the Archimedean approach approximates not just the position of the curve but also its direction.

The rectilinear example shows that some information above and beyond just the position is necessary; we know that directional information is sufficient, in essence, because we can define the arc length using the directional information (more specifically, using the derivatives of the curve): if $\mathbf{f}(t) = \langle f_x(t), f_y(t)\rangle$ is a planar curve, then the arc length of a segment from $t=a$ to $t=b$ is given by the integral $$L=\int_a^b\left|\mathbf{f}'(t)\right|dt = \int_a^b\sqrt{\left(\frac{df_x(t)}{dt}\right)^2+\left(\frac{df_y(t)}{dt}\right)^2}\ dt$$

which depends on the direction of the curve, but also clearly only depends on that information.

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This is related to the approximation of a smooth curve (here, a circle) with its secant. Basically, take a "nicely-behaved" curve (so that the curvature is bounded; in case of a circle, the curvature is constant), and take a sequence of points along this curve that are close enough to each other. Then, the curve length is well approximated by the sum of length of the line segments connecting the points.

Without going into differential calculus, this is explained as zooming in on the curve, it becomes more and more like a straight line. This is why people used to think the Earth is flat: locally, at small distances, the surface can be approximated by a plane.

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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 $n \to \infty.$

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Then how do we make sure that the limit is π ?

That animation simply shows that $\pi<4$, whereas Archimedes put $\pi$ between two limits.

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