# How does $a² + b² = c²$ work with 'steps'? [duplicate]

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 purpleJul 4 '13 at 18:27

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.