The shape in the picture was created by cutting out a small cross from a large cross. Each of these crosses can be composed of five identical squares, with the sides of the small squares being half the length of the sides of the large squares. The area of the gray shape in the picture is $45 \mathrm{~cm}^{2}$.

What is the area of the large cross?

(L. Růžičková)

![](https://cdn.mathpix.com/cropped/2024_04_17_ebb0cc8ff65c04bed773g-3.jpg?height=519&width=531&top_left_y=1802&top_left_x=771)

Hint. How many times does the small square fit into the large one?
The side length of the large square is twice the side length of the small square, so the area of the large square is four times the area of the small square, see the figure. If we remove the small square from the large square, the remaining shape has an area of $\frac{3}{4}$ of the large square. Therefore, the area of the large square is $\frac{4}{3}$ of the remaining shape.

![](https://cdn.mathpix.com/cropped/2024_04_17_ebb0cc8ff65c04bed773g-4.jpg?height=251&width=257&top_left_y=454&top_left_x=905)

The shape in the problem is formed by removing five small squares from five large squares (the exact placement of the squares is not significant for determining the areas). The relationships between their areas are therefore the same as in the discussed case. The area of the large cross is

$$
\frac{4}{3} \cdot 45 = 60 \left(\mathrm{~cm}^{2}\right)
$$

Note. If $S$ represents the area of the small square, then the area of the small cross is $5S$, the area of the large square is $4S$, and the area of the large cross is $20S$. The area of the cut-out shape is $(20-5)S = 15S = 45 \mathrm{~cm}^{2}$, so $S = 3 \mathrm{~cm}^{2}$ and the area of the large cross is $20 \cdot 3 = 60 \left(\mathrm{~cm}^{2}\right)$.