6. From rectangle $A B C D$, a square $A E F G$ has been cut out, such that the area of the resulting polygon is $216 \mathrm{~cm}^{2}$. If $|\overline{E B}|=0.8 \mathrm{dm},|\overline{G D}|=6 \mathrm{~cm}$, what is the area of square $A E F G$?

![](https://cdn.mathpix.com/cropped/2024_06_03_a89fbb865052a7f27847g-10.jpg?height=449&width=679&top_left_y=598&top_left_x=517)
6. 

![](https://cdn.mathpix.com/cropped/2024_05_30_7e3b3cfdf053042fa10ag-11.jpg?height=374&width=560&top_left_y=338&top_left_x=291)

Let the side length of the square AEFG be $a$, the area of the square $A E F G$ be $P$, and the area of the rectangle $A B C D$ be $P_{1}$.

According to the conditions of the problem, we can write:

$P_{1}-P=216$

2 POINTS

$(a+8) \cdot(a+6)-a^{2}=216$

2 POINTS

$a^{2}+8 a+6 a+48-a^{2}=216$

$14 a+48=216$

2 POINTS

$14 a=168$

$a=12 \text{ cm}$,

2 POINTS

Finally, $P=144 \text{ cm}^{2}$