To solve the problem of finding the area of the major sector of a circle that subtends an angle of 120° at the center, we will follow these steps: ### Step 1: Understand the Circle's Properties 1. **Diameter of the Circle**: The diameter is given as \(4\sqrt{3}\) cm. 2. **Radius of the Circle**: The radius \(r\) is half of the diameter: \[ r = \frac{4\sqrt{3}}{2} = 2\sqrt{3} \text{ cm} \] ### Step 2: Calculate the Area of the Circle The area \(A\) of a circle is calculated using the formula: \[ A = \pi r^2 \] Substituting the radius we found: \[ A = \pi (2\sqrt{3})^2 = \pi (4 \cdot 3) = 12\pi \text{ cm}^2 \] ### Step 3: Calculate the Area of the Sector The area of a sector of a circle can be calculated using the formula: \[ \text{Area of Sector} = \frac{\theta}{360^\circ} \times A \] where \(\theta\) is the angle in degrees. Here, \(\theta = 120^\circ\): \[ \text{Area of Sector} = \frac{120}{360} \times 12\pi = \frac{1}{3} \times 12\pi = 4\pi \text{ cm}^2 \] ### Step 4: Calculate the Area of the Major Sector The major sector is the remaining part of the circle after removing the area of the minor sector. Since the total area of the circle is \(12\pi\) cm², we can find the area of the major sector by subtracting the area of the minor sector from the total area: \[ \text{Area of Major Sector} = \text{Total Area} - \text{Area of Minor Sector} \] \[ \text{Area of Major Sector} = 12\pi - 4\pi = 8\pi \text{ cm}^2 \] ### Conclusion: Final Answer The area of the major sector is: \[ \boxed{8\pi} \text{ cm}^2 \] ### Explanation of Other Options - **Option A (4π cm²)**: This is the area of the minor sector, not the major sector. - **Option B (32π cm²)**: This value is not relevant to the problem as it exceeds the total area of the circle. - **Option C (16π cm²)**: This value is also incorrect as it does not correspond to any calculated area related to the sectors of the circle. ### Revision Summary - The radius of the circle is half the diameter. - The area of a sector is calculated using the angle and the total area of the circle. - The area of the major sector is found by subtracting the area of the minor sector from the total area of the circle. - The correct answer for the area of the major sector is \(8\pi\) cm².