Question 9

We need to find the ratio of the area of a square inscribed in a circle to the area of the circle.

Step 1 — Understand the setup

Let the radius of the circle be $r$. The square is inscribed in the circle. This means the vertices of the square touch the circle. The diagonal of the square is the diameter of the circle.

$\text{Diameter of circle} = 2r$

$\text{Diagonal of square} = 2r$

Step 2 — Find the side of the square

Let the side of the square be $a$. We can use the Pythagorean theorem. Consider a right-angled triangle formed by two sides and the diagonal.

$a^2 + a^2 = (\text{diagonal})^2$

$2a^2 = (2r)^2$

$2a^2 = 4r^2$

$a^2 = \frac{4r^2}{2}$

$\boxed{a^2 = 2r^2}$

Step 3 — Calculate the areas

The area of the square is $a^2$. From Step 2, we know $a^2 = \mathbf{2r^2}$.

$\text{Area of square} = 2r^2$

The area of the circle is given by the formula.

$\text{Area of circle} = \pi r^2$

Step 4 — Find the ratio

We need the ratio of the area of the square to the area of the circle.

$\text{Ratio} = \frac{\text{Area of square}}{\text{Area of circle}}$

$\text{Ratio} = \frac{2r^2}{\pi r^2}$

$\text{Ratio} = \frac{2}{\pi}$

Now, let's approximate this value. We know that $\pi \approx \mathbf{3.14159}$.

$\text{Ratio} \approx \frac{2}{3.14159}$

$\text{Ratio} \approx 0.636619$

Rounding to three decimal places, we get $\mathbf{0.637}$.

$\boxed{\text{Ratio} \approx 0.637}$

Answer

The ratio of the area of the square to the area of the circle is $\frac{2}{\pi}$. This ratio is approximately $\mathbf{0.637}$.