Two identical circles each of radius 4 cm intersect 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. `(16pi)/3-4sqrt3`
B. `(32pi)/3-4sqrt3`
C. `(32pi)/3-8sqrt3`
D. `(64pi)/3-16sqrt3`

Answer

To find the area of the intersecting region of two identical circles each of radius 4 cm, we can follow these steps:

### Step-by-step Solution:

1. **Understanding the Configuration**:
- We have two identical circles with a radius of 4 cm.
- The circles intersect such that the circumference of each circle passes through the center of the other.

2. **Finding the Central Angle**:
- The distance between the centers of the two circles is equal to the radius (4 cm).
- The triangle formed by the centers of the circles and one of the intersection points is an equilateral triangle, where each side is equal to the radius (4 cm).
- Therefore, the angle subtended at the center of each circle by the intersection points is 60 degrees.

3. **Calculating the Area of the Sector**:
- The area of a sector of a circle is given by the formula:
\[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \]
- For our circles, \( \theta = 60^\circ \) and \( r = 4 \) cm.
- Thus, the area of one sector is:
\[ \text{Area of sector} = \frac{60}{360} \times \pi \times (4^2) = \frac{1}{6} \times \pi \times 16 = \frac{8\pi}{3} \text{ cm}^2 \]

4. **Calculating the Area of the Equilateral Triangle**:
- The area of an equilateral triangle with side length \( a \) is given by:
\[ \text{Area} = \frac{\sqrt{3}}{4} a^2 \]
- Here, \( a = 4 \) cm.
- Therefore, the area of the triangle is:
\[ \text{Area} = \frac{\sqrt{3}}{4} \times (4^2) = \frac{\sqrt{3}}{4} \times 16 = 4\sqrt{3} \text{ cm}^2 \]

5. **Finding the Area of the Intersecting Region**:
- The area of the intersecting region (shaded area) is the area of the two sectors minus the area of the triangle:
\[ \text{Area of intersecting region} = 2 \times \text{Area of sector} - \text{Area of triangle} \]
- Substituting the values:
\[ \text{Area of intersecting region} = 2 \times \frac{8\pi}{3} - 4\sqrt{3} = \frac{16\pi}{3} - 4\sqrt{3} \text{ cm}^2 \]

### Final Answer:
The area of the intersecting region is:
\[ \frac{16\pi}{3} - 4\sqrt{3} \text{ cm}^2 \]