3. In a circular sector with a central angle of $90^{\circ}$, a square is inscribed such that one of its vertices lies on the arc of the given sector. Express the ratio of the area of the square to the area of the circular sector as a percentage. Use the approximate value $\pi \approx 3.14$ for the number $\pi$. Round the percentage to the nearest whole number.
3. 

![](https://cdn.mathpix.com/cropped/2024_05_30_64eca78245e6b7f11e3dg-2.jpg?height=434&width=437&top_left_y=288&top_left_x=273)

$a$ - length of the side of the square

$d$ - length of the diagonal of the square

$r$ - length of the radius

$d=r$

1 POINT

Apply the Pythagorean theorem to the square:

$a^{2}+a^{2}=r^{2}$,

$a=\frac{\sqrt{2}}{2} r$.

1 POINT

Area of the square:

$P_{k}=a^{2}=\left(\frac{\sqrt{2}}{2} r\right)^{2}=\frac{1}{2} r^{2}$.

Area of the circular sector with a central angle of $90^{\circ}$ is one-fourth of the area of the circle with radius $r$: $P_{i}=\frac{1}{4} r^{2} \pi$.

1 POINT

The ratio of the area of the square to the area of the circular sector is

$P_{k}: P_{i}=\left(\frac{1}{2} r^{2}\right):\left(\frac{1}{4} r^{2} \pi\right)=\frac{2}{\pi}$.

1 POINT

$2: \pi \approx 2: 3.14=0.636 \ldots$

The area of the square is approximately $64\%$ of the area of the circular sector.

1 POINT

. TOTAL 6 POINTS

Note: The area of the circular sector can also be calculated using the central angle of $90^{\circ}$.

$$
P_{i}=\frac{r^{2} \pi \alpha}{360^{\circ}}=\frac{r^{2} \pi \cdot 90^{\circ}}{360^{\circ}}=\frac{1}{4} r^{2} \pi
$$