## Task 2 - 210822

Given the side length $a$ of a square $A B C D$. Around $B$ and $D$, circular arcs with radius $a$ are drawn, enclosing a leaf-like figure (in the image, blue) within the square $A B C D$.

a) Calculate the area of the shaded region for $a=3.5 \mathrm{~cm}$!

b) Determine a general formula that indicates how the area of the leaf-like figure depends on the given side length $a$!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0724.jpg?height=331&width=282&top_left_y=1619&top_left_x=1458)
a) The diagonal $A C$ bisects the square and, due to the symmetrical position of the two circular arcs, also bisects the shaded area.

Therefore, half of the desired area is obtained by subtracting the area of triangle $A B C$ (i.e., half the area of the square) from the area of the quarter circle centered at $B$. Thus, the desired area is

$$
2 \cdot\left(\frac{\pi}{4} \cdot 3.5^{2}-\frac{1}{2} \cdot 3.5^{2}\right) \mathrm{cm}^{2}=\left(\frac{\pi}{2}-1\right) \cdot 3.5^{2} \mathrm{~cm}^{2} \approx 6.98 \mathrm{~cm}^{2}
$$

b) With the same reasoning as in a), the formula for the desired area is

$$
2 \cdot\left(\frac{\pi}{4} \cdot a^{2}-\frac{1}{2} \cdot a^{2}\right)=\left(\frac{\pi}{2}-1\right) a^{2}
$$