Modeling the relationship between perimeter and area Is there any equation that models the relationship between the area and perimeter of a rectangle?
 A: Depending on the elements of the rectangle that you know, you can have different relations. E.g., If you know a side $a$ than: $a(p/2-a)=A$ where $p$ is the perimeter and $A$ the area.
If you know the diagonal $d$ you have: $\left(\dfrac{p}{2}\right)^2-2A=d^2$
A: The area is $A = ab$, while the perimeter is $P= 2(a+b)$. An exact relationship is of course impossible, but you can work out some inequalities. An example is shown below
$(\sqrt{a}-\sqrt{b})^2 \geq 0$ implies that
$a + b \geq 2\sqrt{ab}$, therefore $$(P/4)^2 \geq A$$
I don't know how useful this can be, I guess it depends on the kind of problem you're working on.
A: As pointed out, specific answer is not possible. It depends on what further information is given. The following is another piece of information one can consider. If it is known, the relation between p and A can be setup.
Let the rectangle be a x b with $a$ being the shorter side. It is always possible to have $b = ka$ for some constant $k \ge 1$. Then
$2(a + ka) = p$ which means $a = \frac {p}{2(k+ 1)}$
$A = ab = ka^2 = ... = \frac {kp^2}{4(k + 1)^2}$
Hence, we can say that $A$ is directly proportional to $p^2$ and the proportional constant is $\frac {k}{4(k + 1)^2}$ where k is the aspect ratio of that rectangle.
