Does a Rectangle's Area Always Increase with Its Perimeter?

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SUMMARY

The discussion clarifies that a rectangle's area does not necessarily increase with its perimeter. A specific example is provided using a square with side length "a" and a degenerate rectangle with two sides measuring 3a and the other two sides measuring 0, resulting in a perimeter of 6a but an area of 0. This illustrates that increasing perimeter can lead to a decrease in area, particularly in cases where one dimension approaches zero.

PREREQUISITES
  • Understanding of basic geometry concepts, particularly rectangles and squares.
  • Familiarity with perimeter and area calculations.
  • Basic knowledge of limits and parameterization in mathematics.
  • Introductory calculus concepts, though not strictly necessary for this discussion.
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  • Explore the relationship between perimeter and area in various geometric shapes.
  • Investigate the concept of degenerate shapes in geometry.
  • Learn about parameterization in calculus and its applications in geometry.
  • Study the implications of limits in geometric contexts, particularly with respect to area and perimeter.
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Mathematicians, geometry enthusiasts, educators, and students seeking to deepen their understanding of the relationship between area and perimeter in rectangles and other shapes.

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If the perimeter of a rectangle increases, does the area necessarily increase?

Can anyone explain this using calculus?
 
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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..
 
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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