What is the area of the region of the circle which is situated outside the incribed square of side x ?

Options

A. `(pi-2)x^(2)`
B. `(pi-2)x^(2)//2`
C. `2(pi-2)x^(2)`
D. `(pi-2)x^(2)//4`

Answer

To find the area of the region of the circle that is situated outside the inscribed square of side \( x \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Area of the Square**:
The area \( A_s \) of a square is given by the formula:
\[ A_s = \text{side}^2 = x^2 \]

2. **Determine the Diameter of the Circle**:
The square is inscribed in the circle, which means the diagonal of the square is the diameter of the circle. The diagonal \( d \) of a square can be calculated using the formula:
\[ d = x\sqrt{2} \]

3. **Calculate the Radius of the Circle**:
The radius \( r \) of the circle is half of the diameter:
\[ r = \frac{d}{2} = \frac{x\sqrt{2}}{2} = \frac{x}{\sqrt{2}} \]

4. **Find the Area of the Circle**:
The area \( A_c \) of a circle is given by the formula:
\[ A_c = \pi r^2 \]
Substituting the value of \( r \):
\[ A_c = \pi \left(\frac{x}{\sqrt{2}}\right)^2 = \pi \frac{x^2}{2} \]

5. **Calculate the Area Outside the Square**:
To find the area of the region of the circle that is outside the square, we subtract the area of the square from the area of the circle:
\[ A_{\text{outside}} = A_c - A_s = \pi \frac{x^2}{2} - x^2 \]

6. **Simplify the Expression**:
To combine the terms, we can factor out \( x^2 \):
\[ A_{\text{outside}} = x^2 \left(\frac{\pi}{2} - 1\right) \]

### Final Result:
The area of the region of the circle which is situated outside the inscribed square of side \( x \) is:
\[ A_{\text{outside}} = x^2 \left(\frac{\pi}{2} - 1\right) \]