calculus - How does $a^2 + b^2 = c^2$ work with ‘steps’?

We all know that $a^2+b^2=c^2$ in a right-angled triangle, and therefore, that $c, 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?

Answer

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.