Side of a rectangle

The side of a rectangle can be calculated by rearranging the formula for computing area or perimeter given the other side length. Hence we have the following formulas for computing side $a$ given side $b$.\begin{aligned} A &= ab \quad &&\Rightarrow \quad a &&= \frac{A}{b} \\ p &= 2(a+b) \quad &&\Rightarrow \quad a &&= \frac{p - 2b}{2} \end{aligned}The formulas for computing side $b$ given $a$ are symmetric.

If we have only area and perimeter given of the rectangle and want to compute one of the sides, for instance side $a$, we possibly have two solutions. We rearrange the formulas\begin{aligned} A &= ab \quad &&\Rightarrow \quad b &&= \frac{A}{a} \\ p &= 2(a+b) \quad &&\Rightarrow \quad p &&= 2\left( a+ \frac{A}{a} \right) \end{aligned}by plugging in $b$ from the area formula into the perimeter formula. This then leads to the quadratic equation$a^2 - \frac{p}{2}\cdot a + A = 0$for which we can have two solutions.