ABC is a an equilateral triangle of side 2 cm . With A , B , C as centres and radius 1 cm three arcs are drawn . The area of the region within the triangle bounded by the three arcs is

Options

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

Answer

To find the area of the region within the equilateral triangle ABC bounded by the three arcs, we can follow these steps:

### Step 1: Understand the Triangle
Given that ABC is an equilateral triangle with each side measuring 2 cm, we can denote the vertices as A, B, and C.

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

### Step 3: Determine the Radius and Angle for the Arcs
Each arc is drawn with centers at A, B, and C, and has a radius of 1 cm. The angle subtended by each arc at the center of the triangle is 60 degrees (since the triangle is equilateral).

### Step 4: Calculate the Area of One Arc
The area of a sector of a circle is given by:
\[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \]
For one arc:
- \(\theta = 60\) degrees
- \(r = 1\) cm

So the area of one arc is:
\[ \text{Area of one arc} = \frac{60}{360} \times \pi \times (1)^2 = \frac{1}{6} \pi \text{ cm}^2 \]

### Step 5: Calculate the Total Area of the Three Arcs
Since there are three arcs, the total area of the arcs is:
\[ \text{Total area of arcs} = 3 \times \frac{1}{6} \pi = \frac{1}{2} \pi \text{ cm}^2 \]

### Step 6: Find the Area Bounded by the Arcs
To find the area within the triangle that is bounded by the arcs, we subtract the area of the arcs from the area of the triangle:
\[ \text{Required Area} = \text{Area of triangle} - \text{Total area of arcs} \]
Substituting the values:
\[ \text{Required Area} = \sqrt{3} - \frac{1}{2} \pi \text{ cm}^2 \]

### Final Answer
The area of the region within the triangle bounded by the three arcs is:
\[ \sqrt{3} - \frac{1}{2} \pi \text{ cm}^2 \]