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Calculus and simple geometry

  1. Nov 30, 2004 #1
    If the perimeter of a rectangle increases, does the area necessarily increase?

    Can anyone explain this using calculus?
  2. jcsd
  3. Nov 30, 2004 #2


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    No it does not, and you don't need calculus to find the answer:
    Let's say you've got a square initially with side "a".
    Then, you look at the degenerate rectangle with two sides 3a, and the other two length 0.

    The perimeter of the degenerate rectangle is 3a+0+3a+0=6a, that is, greater than your original square's 4a, yet the rectangle's area is zero..
  4. Nov 30, 2004 #3


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    Think of a rectangle with corners at (0,0), (0,1/x), (x,1/x), (x,0). Draw a picture-one corner at the origin the opposite on the graph 1/x.

    Now we get an entire family of rectangles parameterized by x>0. What can you say about the area of these guys? What about the perimiter as x varies?
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