The rhombus

Define the dimensions of a square
Inscribe a rhombus whose vertices touch the midpoint of each side of the square, and indicate the size of the rhombus
Indicate the area of the square
Indicate the area of the rhombus

A square is defined with its side $l=6$ cm.
Note that the axes of the inscribed rhombus ($D$ and $d$) measure the same as the side of the square $l$. $$D=6 \ \mbox{cm}$$ $$d=6 \ \mbox{cm}$$
The area of the square is: $$A=(6 \ \mbox{cm})^2=36 \ \mbox{cm}^2$$
The area of the rhombus is: $$A_{rhombus}= \dfrac{D\cdot d}{2}=18 \ \mbox{cm}^2 = \dfrac{A_{square}}{2} $$

See that the inscribed rhombus is also a square of side $\sqrt{18}=3\sqrt{2}.$

Thus, the side of the rhombus could also have been calculated (with the Pythagorean theorem) and then squared to obtain the area.