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We all know that $a²+b²=c²$ in a right-angled triangle, and therefore, that $c<a+b$, so that walking along the red line would be shorter than using the two black lines to get from top left to bottom right in the following graphic:

Now, let's assume that the direct way using the red line is blocked, but instead, we can use the green way in the following picture:

Obviously, the green way isn't any shorter than the black one, it's just $a/2+b/2+a/2+b/2 = a+b$. Now, we can divide the green path again, just like the black path, and get to the purple path. Dividing this one in two halfs again, we get the yellow path:

Now obviously, the yellow path is still as long as the black path from the beginning, it's just $8*a/8+8*b/8=a+b$. But if we do this segmentation again and again, we approximate the red line - without making the way any shorter. Why is this so?

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marked as duplicate by vadim123, Zev Chonoles, Inquest, Calvin Lin, Start wearing purple Jul 4 '13 at 18:27

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up vote 3 down vote accepted

Essentially, it is because the distance of the stepped curve from the line does not get small compared to the length of the steps.

An example where the limit is properly found is dividing a circle into $n$ equal parts and computing the sum of the line segments connecting the endpoints of the arcs. This $does$ converge to the length of the circle because the height of each arc gets arbitrarily small compared to the length of each arc as $n$ gets large.

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