in a circle of radius 14 cm , an arc subtends an angle of 120 degrees at the centre. Find the area of the major sector.

To find the area of the major sector, we'll follow these steps:

1. **Calculate the area of the entire circle**.
2. **Calculate the area of the minor sector** (corresponding to the given central angle of 120 degrees).
3. **Subtract the area of the minor sector from the area of the entire circle** to get the area of the major sector.

### Step 1: Area of the Circle
The formula for the area of a circle is:
\[
\text{Area} = \pi r^2
\]
where \( r \) is the radius of the circle. Here, the radius \( r = 14 \) cm.

\[
\text{Area} = \pi \times 14^2 = 196\pi \, \text{cm}^2
\]

### Step 2: Area of the Minor Sector
The formula for the area of a sector is:
\[
\text{Area of sector} = \frac{\theta}{360} \times \pi r^2
\]
where \( \theta \) is the central angle in degrees. For the minor sector, \( \theta = 120 \) degrees.

\[
\text{Area of minor sector} = \frac{120}{360} \times \pi \times 14^2 = \frac{1}{3} \times 196\pi = \frac{196\pi}{3} \, \text{cm}^2
\]

### Step 3: Area of the Major Sector
The major sector is the remaining part of the circle after removing the minor sector. The area of the major sector is given by:

\[
\text{Area of major sector} = \text{Total area of circle} - \text{Area of minor sector}
\]

\[
\text{Area of major sector} = 196\pi - \frac{196\pi}{3}
\]

Simplifying:

\[
\text{Area of major sector} = \frac{3 \times 196\pi}{3} - \frac{196\pi}{3} = \frac{588\pi}{3} - \frac{196\pi}{3} = \frac{392\pi}{3} \, \text{cm}^2
\]

Thus, the area of the major sector is:

\[
\text{Area of major sector} = \frac{392\pi}{3} \approx 410.2 \, \text{cm}^2
\]

Would you like more details on any part of the solution or have any questions?

### Relative Questions:
1. How do you find the area of a segment in a circle?
2. What is the relationship between the central angle and the arc length of a circle?
3. How do you calculate the length of an arc?
4. Can you explain the difference between a sector and a segment of a circle?
5. How do you find the perimeter of a sector?

**Tip:** Always remember that the area of a sector is proportional to the central angle. So, the fraction \(\frac{\theta}{360}\) is crucial when calculating sector areas.

In a circle of radius 14 cm, an arc subtends an angle of 120 degrees at the center. Find the area of the major sector.

Learn how to calculate the area of a major sector in a circle with a radius of 14 cm and a central angle of 120 degrees. The problem involves using geometry formulas such as the area of a circle and the area of a sector. Detailed steps included, suitable for students in Grades 9-10.

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