Three coins of the same size (radius 1 cm) are placed on a table such that each of them touches the other two. The area enclosed by the coins is

Options

A. (a) `((pi)/(2) - sqrt(3)) cm^(2)`
B. (b) `(sqrt(3) - (pi)/(2))cm^(2)`
C. (c) `(2sqrt(3) - (pi)/(2)) cm^(2)`
D. (d) `(3sqrt(3) - (pi)/(2)) cm^(2)`

Answer

To find the area enclosed by the three coins, we can follow these steps:

### Step 1: Understand the Configuration
We have three coins, each with a radius of 1 cm, placed such that each coin touches the other two. This arrangement forms an equilateral triangle with the centers of the coins as the vertices.

### Step 2: Calculate the Side Length of the Triangle
Since each coin has a radius of 1 cm, the distance between the centers of any two coins (which forms the side of the triangle) is:
\[ \text{Side length} = 1 \text{ cm} + 1 \text{ cm} = 2 \text{ cm} \]

### Step 3: Calculate the Area of the Equilateral Triangle
The formula for the area \( A \) of an equilateral triangle with side length \( s \) is:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
Substituting \( s = 2 \) cm:
\[ A = \frac{\sqrt{3}}{4} \times (2)^2 = \frac{\sqrt{3}}{4} \times 4 = \sqrt{3} \text{ cm}^2 \]

### Step 4: Calculate the Area of the Circular Sectors
Each coin contributes a sector to the enclosed area. The angle of the sector corresponding to each coin is \( 60^\circ \) (since the triangle is equilateral). The area \( A_s \) of one sector is given by:
\[ A_s = \frac{\theta}{360^\circ} \times \pi r^2 \]
where \( \theta = 60^\circ \) and \( r = 1 \) cm. Thus:
\[ A_s = \frac{60}{360} \times \pi \times (1)^2 = \frac{1}{6} \pi \text{ cm}^2 \]
Since there are three sectors:
\[ \text{Total area of sectors} = 3 \times A_s = 3 \times \frac{1}{6} \pi = \frac{1}{2} \pi \text{ cm}^2 \]

### Step 5: Calculate the Enclosed Area
The area enclosed by the coins is the area of the triangle minus the total area of the sectors:
\[ \text{Enclosed Area} = \text{Area of Triangle} - \text{Total Area of Sectors} \]
Substituting the values we calculated:
\[ \text{Enclosed Area} = \sqrt{3} - \frac{1}{2} \pi \text{ cm}^2 \]

### Conclusion
The area enclosed by the coins is:
\[ \sqrt{3} - \frac{\pi}{2} \text{ cm}^2 \]

### Final Answer
The correct option is \( \sqrt{3} - \frac{\pi}{2} \text{ cm}^2 \).

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