What is the Relationship Between Perimeter and Area in an Infinite Staircase?

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Alright, so I was just browsing 4Chan and I came across this post.

[PLAIN]http://img121.imageshack.us/img121/5374/1291537737867.jpg

I realize infinitely making corners out of corners may result in an approximation of a perfect curve, however, it will always be jagged and thus result in the difference between 4 and pi.

My question is, can you maintain the same perimeter while infinitely reducing the area? Cause that's what it appears to be doing in the picture.
 
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The perimeter will polygon will always equal 4. The area, of the polygon, however, will come infinitely close to the area of the circle. At least that's my understanding.
 
That's a variation on the problem where you take more and more "stairsteps" from (0, 0) to (1, 1) getting a figure very close to the straight line from (0, 0) to (1, 1) but showing that the total length is always "2", not the length of the straight line, [math]\sqrt{2}[/math]. Essentially, the problem is that the stairsteps do not converge uniformly to the line.
 
HallsofIvy said:
Essentially, the problem is that the stairsteps do not converge uniformly to the line.

They do converge uniformly to the line.