13. Lucio buys a table in the shape of a regular hexagon and a rectangular tablecloth with an area of $1024 \mathrm{~cm}^{2}$ that has the shorter side equal to the side of the hexagon and covers exactly the portion of the table between two opposite sides. What is the area of the table, expressed in $\mathrm{cm}^{2}$?
13. The answer is 1536.

![](https://cdn.mathpix.com/cropped/2024_04_17_9f6e2f6c8f4b083321c1g-12.jpg?height=728&width=825&top_left_y=1418&top_left_x=661)

Let $A B C D E F$ be a regular hexagon representing the table; the tablecloth with an area of $1024 \mathrm{~cm}^{2}$ exactly covers the rectangle $A B D E$. Let $O$ be the center of the hexagon, and let $H, K$ be the points of intersection between the segments $B D$ and $O C, A E$ and $O F$, respectively. The triangles $O A K, O E K, O B H$, and $O D H$ are all congruent and have an area equal to half the area of the equilateral triangles $A B O$ and $D E O$. It follows that the area of the rectangle $A B D E$ is four times the area of $A B O$, while the area of the table is six times the area of $A B O$. Therefore, the area of the table is $\frac{6}{4} \cdot 1024 \mathrm{~cm}^{2}=1536 \mathrm{~cm}^{2}$.