Three circles of raidus 21 cm 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. `441sqrt(3) - 693`
B. `882sqrt(3) - 693`
C. `882sqrt(3) - 462`
D. `441sqrt(3) - 462`

Answer

To find the area enclosed by three circles of radius 21 cm that touch each other, we can follow these steps:

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

### Step 2: Calculate the Side Length of the Triangle
Since each circle has a radius of 21 cm, the distance between the centers of any two circles (which is the side length of the equilateral triangle) is:
\[ \text{Side length} = 21 \, \text{cm} + 21 \, \text{cm} = 42 \, \text{cm} \]

### Step 3: Calculate the Area of the Equilateral Triangle
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} \times a^2 \]
where \( a \) is the side length. Substituting \( a = 42 \, \text{cm} \):
\[ A = \frac{\sqrt{3}}{4} \times (42)^2 \]
\[ A = \frac{\sqrt{3}}{4} \times 1764 \]
\[ A = 441\sqrt{3} \, \text{cm}^2 \]

### Step 4: Calculate the Area of One Sector
Each circle contributes a sector to the enclosed area. The angle of each sector is \( 60^\circ \) (since the triangle is equilateral). The area of a sector is given by:
\[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \]
where \( \theta = 60^\circ \) and \( r = 21 \, \text{cm} \):
\[ \text{Area of sector} = \frac{60}{360} \times \pi \times (21)^2 \]
\[ = \frac{1}{6} \times \pi \times 441 \]
\[ = \frac{441\pi}{6} \, \text{cm}^2 \]

### Step 5: Calculate the Total Area of the Three Sectors
Since there are three circles, the total area of the three sectors is:
\[ \text{Total area of sectors} = 3 \times \frac{441\pi}{6} \]
\[ = \frac{1323\pi}{6} \, \text{cm}^2 \]

### Step 6: Calculate the Enclosed Area
The area enclosed by the three circles is the area of the triangle minus the total area of the three sectors:
\[ \text{Enclosed Area} = \text{Area of triangle} - \text{Total area of sectors} \]
\[ = 441\sqrt{3} - \frac{1323\pi}{6} \, \text{cm}^2 \]

### Final Answer
Thus, the area of the portion enclosed by the three circles is:
\[ \text{Enclosed Area} = 441\sqrt{3} - \frac{1323\pi}{6} \, \text{cm}^2 \]

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