A mosaic tile pattern is shown in the figure. The centers of the arcs are at the vertices of the square; two of the arcs touch each other. What fraction of the area of the square is the hatched area?

![](https://cdn.mathpix.com/cropped/2024_05_02_9495ee0295e47705d6c1g-1.jpg?height=350&width=358&top_left_y=264&top_left_x=878)
Let's choose the side of the square to be of unit length. Then the area $t_{1}$ of the rounded sector is the difference between the area of a quarter circle with radius 1 and the area of a right-angled triangle with a leg of length 1, i.e., $t_{1}=\frac{\pi}{4}-\frac{1}{2}$.

The radius of the small quarter circle is $r_{1}=\sqrt{2}-1$, and its area is $t_{2}=\frac{(\sqrt{2}-1)^{2} \pi}{4}$. Due to the diagonal symmetry of the square, each area appears twice, so the area of the rounded part is shown in Figure 1:

$$
2\left(t_{1}+t_{2}\right)=\frac{\pi-2}{2}+\frac{(\sqrt{2}-1)^{2} \pi}{2}=(2-\sqrt{2}) \pi-1 \approx 0.84029
$$

which is approximately $8/10$ of the area of the square.

Remark. The values of $\sqrt{2}$ and $\pi$ were calculated to 4 decimal places. Most solvers obtained a result of 0.8526. The difference arises because they used only 2 decimal places. Most solvers, instead of simplifying the expression to its simplest form, substituted the decimal values of $\sqrt{2}$ and $\pi$ at the beginning. This not only results in extra work but also reduces accuracy. Always strive for simplification!

![](https://cdn.mathpix.com/cropped/2024_05_02_35f62cfc4edd3fcf6c1cg-1.jpg?height=320&width=328&top_left_y=824&top_left_x=888)