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. `(pi/2 - sqrt3) cm^2`
B. `(sqrt3 - pi/2) cm^2`
C. `(2sqrt3 - pi/2) cm^2`
D. `(3sqrt3 -pi/2) cm^2`

Answer

To find the area enclosed by three coins of the same size (radius 1 cm) placed on a table such that each touches the other two, we can follow these steps:

### Step 1: Understand the Configuration
We have three coins, each with a radius of 1 cm, arranged in such a way that they form an equilateral triangle with their centers. The distance between the centers of any two coins is equal to the sum of their radii, which is 1 cm + 1 cm = 2 cm.

### Step 2: Determine the Area of the Triangle
The centers of the coins form an equilateral triangle (let's denote the centers as A, B, and C). The side length of this triangle is 2 cm. The area \( A_{triangle} \) of an equilateral triangle can be calculated using the formula:
\[ A_{triangle} = \frac{\sqrt{3}}{4} s^2 \]
where \( s \) is the side length. Here, \( s = 2 \) cm.

Calculating the area:
\[ A_{triangle} = \frac{\sqrt{3}}{4} \times (2)^2 = \frac{\sqrt{3}}{4} \times 4 = \sqrt{3} \text{ cm}^2 \]

### Step 3: Calculate the Area of the Sector
Each coin creates a sector in the triangle. Since the angle at each vertex of the triangle is 60 degrees, the area of one sector can be calculated using the formula:
\[ A_{sector} = \frac{\theta}{360} \times \pi r^2 \]
where \( \theta = 60 \) degrees and \( r = 1 \) cm. Thus, the area of one sector is:
\[ A_{sector} = \frac{60}{360} \times \pi \times (1)^2 = \frac{1}{6} \pi \text{ cm}^2 \]

Since there are three sectors (one for each coin), the total area of the sectors is:
\[ A_{sectors} = 3 \times A_{sector} = 3 \times \frac{1}{6} \pi = \frac{1}{2} \pi \text{ cm}^2 \]

### Step 4: Calculate the Enclosed Area
The area enclosed by the coins (the shaded area) is the area of the triangle minus the total area of the sectors:
\[ A_{enclosed} = A_{triangle} - A_{sectors} \]
Substituting the values we calculated:
\[ A_{enclosed} = \sqrt{3} - \frac{1}{2} \pi \text{ cm}^2 \]

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