Two identical circles each of radius 2 cm intersect each other such that the circumference of each one passes through the centre of the other. What is the area (in `cm^(2)`) of the intersecting region ?

Options

A. `(8pi)/(3) -2sqrt(3)`
B. `(8pi)/(3) - sqrt(3)`
C. `(4pi)/(3)-sqrt(3)`
D. `(4pi)/(3) - 2sqrt(3)`

Answer

To find the area of the intersecting region of two identical circles each with a radius of 2 cm, where the circumference of each circle passes through the center of the other, we can follow these steps:

### Step 1: Understand the Geometry
The two circles intersect in such a way that the distance between their centers is equal to the radius of the circles. Since the radius is 2 cm, the distance between the centers of the two circles is also 2 cm.

### Step 2: Calculate the Area of One Circular Segment
To find the area of the intersecting region, we need to calculate the area of one circular segment and then double it (since both segments are identical).

1. **Find the angle (θ) of the circular segment**:
The angle θ (in radians) can be found using the cosine rule in the triangle formed by the centers of the circles and one of the intersection points:
\[ \cos(\theta/2) = \frac{r}{d} = \frac{2}{2} = 1 \]
This means that θ/2 = 60 degrees (or π/3 radians), thus θ = 120 degrees (or 2π/3 radians).

2. **Calculate the area of the sector**:
The area of the sector of one circle is given by:
\[ \text{Area of sector} = \frac{θ}{2\pi} \times \pi r^2 = \frac{2\pi/3}{2\pi} \times \pi (2^2) = \frac{2}{3} \times 4\pi = \frac{8\pi}{3} \text{ cm}^2 \]

3. **Calculate the area of the triangle**:
The area of the triangle formed by the two centers and the intersection point can be calculated using the formula:
\[ \text{Area of triangle} = \frac{1}{2} \times base \times height \]
Here, the base is the distance between the centers (2 cm) and the height can be calculated using the sine of the angle:
\[ \text{Height} = r \sin(\theta/2) = 2 \sin(60^\circ) = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3} \text{ cm} \]
Therefore, the area of the triangle is:
\[ \text{Area of triangle} = \frac{1}{2} \times 2 \times \sqrt{3} = \sqrt{3} \text{ cm}^2 \]

### Step 3: Calculate the Area of One Circular Segment
The area of one circular segment is given by:
\[ \text{Area of circular segment} = \text{Area of sector} - \text{Area of triangle} = \frac{8\pi}{3} - \sqrt{3} \text{ cm}^2 \]

### Step 4: Calculate the Total Area of the Intersecting Region
Since there are two identical segments, the total area of the intersecting region is:
\[ \text{Total area} = 2 \times \left(\frac{8\pi}{3} - \sqrt{3}\right) = \frac{16\pi}{3} - 2\sqrt{3} \text{ cm}^2 \]

### Final Answer
The area of the intersecting region is:
\[ \frac{16\pi}{3} - 2\sqrt{3} \text{ cm}^2 \]