Given a rectangular area, how do I find the largest possible perimeter?

1 Answer
Aug 15, 2014

Trick question; the answer is undefined.

For rectangles, we have:
#A=lw#
#P=2(l+w)#

We can rearrange:
#l=A/w#
Because of symmetry, we want to look for the largest #l#. However, we can make #l# as big as we want by making #w# as small as needed. Therefore #l# goes to infinity which makes #P# go to infinity, so the answer is undefined.

If the question is reversed, where we are looking for the largest area given a perimeter, then we would have a question that has a solution.

Or you can reverse the question to we are looking for smallest perimeter given a rectangular area; this too would have a solution.