Three circles of radius 63 cm each are placed in such a way that each circle touches the other two. What is the area of the portion enclosed by the three circles ? (in sq. cm)

Options

A. a)`7938 sqrt(3) - 4158`
B. b)`3969 sqrt(3) - 4158`
C. c)`7938sqrt(3) - 6237`
D. d)`3969 sqrt(3) - 6237`

Answer

To find the area of the portion enclosed by the three circles, we can follow these steps:

### Step 1: Understand the Configuration
The three circles are arranged such that each circle touches the other two. This forms an equilateral triangle with the centers of the circles as the vertices.

### Step 2: Calculate the Side Length of the Triangle
Since each circle has a radius of 63 cm, the distance between the centers of any two circles (which is the side length of the triangle) is equal to the sum of their radii. Thus, the side length \( s \) of the triangle is:
\[ s = 63 \, \text{cm} + 63 \, \text{cm} = 126 \, \text{cm} \]

### Step 3: Calculate the Area of the Equilateral Triangle
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
Substituting the value of \( s \):
\[ A = \frac{\sqrt{3}}{4} (126)^2 \]
Calculating \( 126^2 \):
\[ 126^2 = 15876 \]
Now substituting this value into the area formula:
\[ A = \frac{\sqrt{3}}{4} \times 15876 \]
Calculating:
\[ A \approx \frac{1.732}{4} \times 15876 \approx 6883.5 \, \text{cm}^2 \]

### Step 4: Calculate the Area of the Circular Segments
Each circle contributes a segment to the enclosed area. The angle subtended at the center of each circle by the triangle is \( 60^\circ \) (since it is an equilateral triangle). The area of the circular segment can be calculated as follows:

1. Area of the sector of the circle:
\[ \text{Area of sector} = \frac{60}{360} \times \pi r^2 = \frac{1}{6} \times \pi (63)^2 \]
Calculating \( (63)^2 \):
\[ (63)^2 = 3969 \]
Thus,
\[ \text{Area of sector} = \frac{1}{6} \times \pi \times 3969 \approx 2090.5 \, \text{cm}^2 \]

2. Area of the triangle formed by the radius and the segment:
The area of the triangle formed by the two radii and the line segment is:
\[ \text{Area of triangle} = \frac{1}{2} \times r \times r \times \sin(60^\circ) = \frac{1}{2} \times 63 \times 63 \times \frac{\sqrt{3}}{2} \]
Calculating:
\[ \text{Area of triangle} = \frac{1}{2} \times 3969 \times \frac{\sqrt{3}}{2} \approx 1715.5 \, \text{cm}^2 \]

3. Area of the circular segment:
\[ \text{Area of segment} = \text{Area of sector} - \text{Area of triangle} \]
\[ \text{Area of segment} \approx 2090.5 - 1715.5 = 375 \, \text{cm}^2 \]

### Step 5: Total Area Enclosed by the Three Circles
Since there are three segments:
\[ \text{Total area enclosed} = \text{Area of triangle} + 3 \times \text{Area of segment} \]
\[ \text{Total area enclosed} \approx 6883.5 + 3 \times 375 = 6883.5 + 1125 = 8008.5 \, \text{cm}^2 \]

### Final Answer
The area of the portion enclosed by the three circles is approximately:
\[ \boxed{8008.5 \, \text{cm}^2} \]