If the four equal circles of radius 3 cm touch each other externally, then the area of the region bounded by the four circles is

Options

A. `4(9-pi)` sq. cm
B. `9(4-pi)` sq. cm.
C. `5(6-pi)` sq. cm.
D. `6(5-pi)` sq. cm

Answer

To find the area of the region bounded by four equal circles of radius 3 cm that touch each other externally, we can follow these steps:

### Step 1: Understand the Configuration
We have four circles, each with a radius of 3 cm, arranged such that they touch each other externally. The centers of these circles form a square.

### Step 2: Determine the Side Length of the Square
The distance between the centers of two adjacent circles is equal to the sum of their radii. Since all circles have the same radius (R = 3 cm), the distance between the centers (side of the square) is:
\[ AB = R + R = 3 + 3 = 6 \text{ cm} \]

### Step 3: Calculate the Area of the Square
The area of the square (ABCD) formed by the centers of the circles can be calculated using the formula for the area of a square:
\[ \text{Area of square} = \text{side}^2 = (6 \text{ cm})^2 = 36 \text{ cm}^2 \]

### Step 4: Calculate the Area of One Circle
The area of one circle can be calculated using the formula:
\[ \text{Area of circle} = \pi R^2 \]
Substituting R = 3 cm:
\[ \text{Area of one circle} = \pi (3)^2 = 9\pi \text{ cm}^2 \]

### Step 5: Calculate the Area of the Four Circles
Since there are four circles, the total area of the four circles is:
\[ \text{Total area of four circles} = 4 \times 9\pi = 36\pi \text{ cm}^2 \]

### Step 6: Calculate the Area of the Bounded Region
The area of the region bounded by the four circles is given by the area of the square minus the area of the four quarter circles that overlap at the corners:
\[ \text{Area of bounded region} = \text{Area of square} - \text{Total area of four circles} \]
The area of the overlapping region (four quarter circles) is equal to the area of one full circle:
\[ \text{Area of bounded region} = 36 - 36\pi \text{ cm}^2 \]

### Step 7: Substitute the Values
Now, substituting the values we have:
\[ \text{Area of bounded region} = 36 - 36\pi \]
This can be rewritten as:
\[ \text{Area of bounded region} = 9 \times 4 - 9\pi \]

### Final Answer
Thus, the area of the region bounded by the four circles is:
\[ \text{Area} = 9(4 - \pi) \text{ cm}^2 \]