Exercises: Square Roots

For the calculation of the side length of the square you need the area $A_{\mathrm{square}}$ of the square. The rectangle and the square have the same area, so you may calculate the area of the rectangle $A_{\mathrm{rectangle}}$, first:

$\begin{array}{rcl}{A}_{\mathrm{rectangle}}& =& \mathrm{length}\cdot \mathrm{width}\\ & & \\ & =& c\cdot b\\ & & \\ & =& 8\mathrm{cm}\cdot 2\mathrm{cm}\\ & & \\ & =& 16{\mathrm{cm}}^2\end{array}$

The area of the square $A_{\mathrm{square}}$ is $16{\mathrm{cm}}^2$. With this information, you may calculate the side length $a$ of the square:

$\begin{array}{rcl}\mathrm{length}\cdot \mathrm{width}& =& A_{\mathrm{square}}\\ & & \\ a\cdot a& =& 16{\mathrm{cm}}^2\\ & & \\ a^2& =& 16{\mathrm{cm}}^2\\ & & \\ \sqrt{a^2}& =& \sqrt{16{\mathrm{cm}}^2}\\ & & \\ a& =& 4\mathrm{cm}\end{array}$

The side lengths of the square thus have a length of $4\mathrm{cm}$.