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Squaring the Circle in the Kitchen

Consider this claim called "Pan Arithmetic," taken from a cookbook found in a "free" box: An 8" square pan holds 1 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.

Note: Question #3 suggested by M.J. (Bellingham).


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

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


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."