A sqaure of side 4 cm is inscirbed in a circular quadrant such that one of the corners of the square coincides with the centre of the quadrant and the diagonally opposite corner lies on the periphery of the quadrant. The area of the quadrant not covered by the square is

Options

A. `8(pi-2)`
B. `4(pi-2)`
C. `8(pi-1)`
D. `4(pi-1)`

Answer

To solve the problem, we need to find the area of the quadrant that is not covered by the square. Here’s a step-by-step solution:

### Step 1: Determine the side length of the square
The side length of the square is given as 4 cm.

### Step 2: Calculate the radius of the circular quadrant
Since one corner of the square coincides with the center of the quadrant and the diagonally opposite corner lies on the periphery, we can use the Pythagorean theorem to find the radius (r) of the quadrant.

The two sides of the square that meet at the center are both 4 cm. Therefore, we can set up the equation:
\[ r^2 = 4^2 + 4^2 \]
\[ r^2 = 16 + 16 \]
\[ r^2 = 32 \]
\[ r = \sqrt{32} = 4\sqrt{2} \, \text{cm} \]

### Step 3: Calculate the area of the quadrant
The area \( A \) of a quadrant is given by the formula:
\[ A = \frac{1}{4} \pi r^2 \]
Substituting the value of \( r^2 \):
\[ A = \frac{1}{4} \pi (32) \]
\[ A = 8\pi \, \text{cm}^2 \]

### Step 4: Calculate the area of the square
The area \( A_s \) of the square is calculated as:
\[ A_s = \text{side}^2 = 4 \times 4 = 16 \, \text{cm}^2 \]

### Step 5: Calculate the area of the quadrant not covered by the square
To find the area of the quadrant that is not covered by the square, we subtract the area of the square from the area of the quadrant:
\[ A_{not \, covered} = A - A_s \]
\[ A_{not \, covered} = 8\pi - 16 \, \text{cm}^2 \]

### Final Answer
Thus, the area of the quadrant not covered by the square is:
\[ 8(\pi - 2) \, \text{cm}^2 \]

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