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

Aug 15, 2014

Trick question; the answer is undefined.

For rectangles, we have:
$A = l w$
$P = 2 \left(l + w\right)$

We can rearrange:
$l = \frac{A}{w}$
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.