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A friend showed me a circle inside a square such that the diameter of the circle is equal to the side length of the square. The perimeter of the square is 8r. He then proceeded to take small squares of the corners of the original square, maintaining perimeter as 8r. As this continues the original square begins to appear like a circle, somehow making it seem like the perimeter of the circle is 8r, which I know isn't true.

What is the flaw in the method above to show that it the perimeter is not 8r?

CipherBot
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  • This is a well known paradox with several explanations on this site that I can't find right now. Someone will. The issue is essentially that approximating a slanted line by a staircase with small steps does not approximate the length of the line. – Ethan Bolker Mar 21 '19 at 12:50
  • The same faulty argument demonstrates that the length of the square’s diagonal is equal to double its side length. Can you work out the argument’s flaw in that case? – amd Mar 21 '19 at 18:37

1 Answers1

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Arclength isn't continuous with respect to uniform convergence.

Umberto P.
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  • “Well, art is art, isn't it? Still, on the other hand, water is water! And east is east and west is west and if you take cranberries and stew them like applesauce they taste much more like prunes than rhubarb does." - Groucho Marx. – Steven Alexis Gregory Mar 21 '19 at 12:55
  • Hooray for Captain Spaulding! – Umberto P. Mar 21 '19 at 12:58