The length of the diagonal of square is 8 cm. A circle has been drawn circumscibing the square. The area of the portion between the circle and the square (in sq cm) is

Options

A. `16 (2)/7`
B. `18 (2)/7`
C. `10 (2)/7`
D. `12 (2)/7`

Answer

To find the area of the portion between the circle and the square, we will follow these steps:

### Step 1: Find the side length of the square
Given the diagonal of the square is 8 cm, we can use the relationship between the diagonal (d) and the side length (s) of a square:
\[ d = s \sqrt{2} \]
So, we can rearrange this to find the side length:
\[ s = \frac{d}{\sqrt{2}} = \frac{8}{\sqrt{2}} = \frac{8 \sqrt{2}}{2} = 4\sqrt{2} \text{ cm} \]

### Step 2: Calculate the area of the square
The area (A) of the square can be calculated using the formula:
\[ A = s^2 \]
Substituting the value of s:
\[ A = (4\sqrt{2})^2 = 16 \times 2 = 32 \text{ cm}^2 \]

### Step 3: Find the radius of the circumscribed circle
The radius (r) of the circle that circumscribes the square is half the length of the diagonal:
\[ r = \frac{d}{2} = \frac{8}{2} = 4 \text{ cm} \]

### Step 4: Calculate the area of the circle
The area (A) of the circle can be calculated using the formula:
\[ A = \pi r^2 \]
Substituting the value of r:
\[ A = \pi (4)^2 = 16\pi \text{ cm}^2 \]
Using \(\pi \approx \frac{22}{7}\):
\[ A \approx 16 \times \frac{22}{7} = \frac{352}{7} \text{ cm}^2 \]

### Step 5: Calculate the area of the portion between the circle and the square
To find the area of the portion between the circle and the square, we subtract the area of the square from the area of the circle:
\[ \text{Area between} = \text{Area of the circle} - \text{Area of the square} \]
Substituting the areas we calculated:
\[ \text{Area between} = \frac{352}{7} - 32 \]
Converting 32 to a fraction with a denominator of 7:
\[ 32 = \frac{224}{7} \]
Now, subtracting:
\[ \text{Area between} = \frac{352}{7} - \frac{224}{7} = \frac{128}{7} \text{ cm}^2 \]

### Step 6: Convert to mixed number
To convert \(\frac{128}{7}\) to a mixed number:
\[ 128 \div 7 = 18 \quad \text{remainder } 2 \]
Thus, \(\frac{128}{7} = 18 \frac{2}{7} \text{ cm}^2\).

### Final Answer:
The area of the portion between the circle and the square is \(18 \frac{2}{7} \text{ cm}^2\).

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