Is there an equation that connect between perimeter and area of a rectangle??

Question

So confused couldn't find an equation that connect between perimeter and area of a rectangle .Some help would be great, thanks

Answer 1

A rectangle's perimeter does not determine its area, and its area does not determine its perimeter.

For example, consider a rectangle whose perimeter is 20. Then its length could be any value $x$ in the interval $0<x<10$ (whether you regard $x=0$ or $x=10$ as legitimate possibilities here depends on whether you want to include two overlapping line segments with no space between them as a degenerate special case of a "rectangle"; this is more a question of semantics than mathematics), and its width would then be $10-x$, so that the area would be equal to $x(10-x)$. This can take on any value between $0$ (imagine a very long, skinny rectangle enclosing almost no area at all) and $25$ (a square).

Since the perimeter does not determine the area, there is no way to have a formula that tells you one in terms of the other. The best you can hope for is an inequality, and indeed $$0 < A \le \left( \frac P4 \right)^2$$ gives you the smallest and largest possible area for a rectangle with perimeter $P$.