The four equal circles of radius 4 cm drawn of the four corners of a square touch each other externally. Then the area of the portion between the square and the four sectors is

Options

A. `9(pi-4)` sq. cm
B. `16(pi-4)` sq. cm
C. `9(4-pi)` sq. cm.
D. `16(4-pi)` sq. cm.

Answer

To solve the problem, we need to find the area of the portion between the square and the four sectors formed by the circles at the corners of the square. Here’s a step-by-step solution:

### Step 1: Determine the side length of the square
Given that the radius of each circle is 4 cm, and since the circles touch each other externally, the distance between the centers of two adjacent circles is equal to the sum of their radii.

Thus, the side length of the square can be calculated as:
\[ \text{Side length of the square} = \text{Radius} \times 2 = 4 \, \text{cm} \times 2 = 8 \, \text{cm} \]

### Step 2: Calculate the area of the square
The area of the square is given by the formula:
\[ \text{Area of the square} = \text{side}^2 = 8 \, \text{cm} \times 8 \, \text{cm} = 64 \, \text{cm}^2 \]

### Step 3: Calculate the area of one circle
The area of a circle is calculated using the formula:
\[ \text{Area of a circle} = \pi r^2 \]
Substituting the radius (4 cm):
\[ \text{Area of one circle} = \pi (4 \, \text{cm})^2 = \pi \times 16 \, \text{cm}^2 = 16\pi \, \text{cm}^2 \]

### Step 4: Calculate the area of the four circles
Since there are four circles, the total area of the circles is:
\[ \text{Total area of four circles} = 4 \times 16\pi \, \text{cm}^2 = 64\pi \, \text{cm}^2 \]

### Step 5: Calculate the area of the sectors
Each circle contributes a quarter of its area to the square. Therefore, the area of the four sectors (one from each circle) is:
\[ \text{Area of four sectors} = 4 \times \left(\frac{1}{4} \times 16\pi \, \text{cm}^2\right) = 16\pi \, \text{cm}^2 \]

### Step 6: Calculate the area between the square and the four sectors
To find the area between the square and the sectors, we subtract the area of the sectors from the area of the square:
\[ \text{Area between square and sectors} = \text{Area of the square} - \text{Area of the four sectors} \]
\[ = 64 \, \text{cm}^2 - 16\pi \, \text{cm}^2 \]

### Final Answer
The area of the portion between the square and the four sectors is:
\[ 64 - 16\pi \, \text{cm}^2 \]