Source: http://mathnexus.wwu.edu/Archive/problem/detail.asp?ID=160
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Consider this claim called "Pan Arithmetic," taken from a
cookbook found in a "free" box: An 8" square pan holds one
cup more than an 8" round one.

If the depth is the same, a square pan holds the same amount
as a round one measuring 1 inch more across.  For example,
an 8" square pan holds the same amount as a 9" round one.

Question 1: Why is the first statement quite incorrect....
Hint: Should it matter what the depth of the pan is?

Question 2: For the second claim, what is the value of pi
being used to make this equality "work"?

Question 3: For what sizes of pans is this estimate good?
Turns out for a square 8" pan, the estimate of a 9" circle
is very close.

Source: Polly Clingerman's The Kitchen Companion, 1994, p. 57

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Hint: Set up equations for area of each pan....and ask,
does the depth of the pan matter?

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Finger pointing down from darrell94590 on 1/2/2006

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Drawing from Ripleys-Believe It Or Not

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              THE SOLUTION
 

Solution Commentary: I will let you play with the first two
questions...and argue over the nuances.

As to Question #3, M.J. (Bellingham) points out: "Graph the
difference between the areas of a square with side 2r and a
circle with radius (r+1).  A zero occurs at approx r is 7.789,
the other zero at about -.47.  Graph is necessarily a parabola
so only close to r = 8 does this approx work and all cooking
pans I have are either 9" square or 8" rounds.  This is
interesting.  Someone planned this out."