Find the area of the minor segment of a circle of radius 14 cm, when its centreal angle is `60^(@)`. Also find the area of the corresponding major segment. `["Use" pi = (22)/(7)]`

Answer

To find the area of the minor segment of a circle with a radius of 14 cm and a central angle of 60 degrees, we can follow these steps:

### Step 1: Calculate the area of the sector
The area of a sector of a circle can be calculated using the formula:
\[ \text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]
where:
- \(\theta\) is the central angle in degrees,
- \(r\) is the radius of the circle.

Given:
- \(\theta = 60^\circ\)
- \(r = 14 \text{ cm}\)
- \(\pi = \frac{22}{7}\)

Substituting the values:
\[ \text{Area of Sector} = \frac{60}{360} \times \frac{22}{7} \times (14)^2 \]

### Step 2: Simplify the calculation
First, simplify \(\frac{60}{360}\):
\[ \frac{60}{360} = \frac{1}{6} \]
Now, calculate \((14)^2\):
\[ (14)^2 = 196 \]
Now substitute back:
\[ \text{Area of Sector} = \frac{1}{6} \times \frac{22}{7} \times 196 \]

### Step 3: Calculate the area of the triangle
The area of the triangle formed by the radius and the chord can be calculated using:
\[ \text{Area of Triangle} = \frac{1}{2} r^2 \sin(\theta) \]
For \(\theta = 60^\circ\), \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\).

Substituting the values:
\[ \text{Area of Triangle} = \frac{1}{2} \times (14)^2 \times \frac{\sqrt{3}}{2} \]
\[ = \frac{1}{2} \times 196 \times \frac{\sqrt{3}}{2} \]
\[ = \frac{196 \sqrt{3}}{4} \]
\[ = 49 \sqrt{3} \]

### Step 4: Calculate the area of the minor segment
The area of the minor segment is given by:
\[ \text{Area of Minor Segment} = \text{Area of Sector} - \text{Area of Triangle} \]
Substituting the values:
\[ \text{Area of Minor Segment} = \left(\frac{1}{6} \times \frac{22}{7} \times 196\right) - (49 \sqrt{3}) \]

### Step 5: Calculate the area of the major segment
The area of the major segment can be found by subtracting the area of the minor segment from the area of the circle:
\[ \text{Area of Circle} = \pi r^2 = \frac{22}{7} \times (14)^2 = \frac{22}{7} \times 196 \]
\[ \text{Area of Major Segment} = \text{Area of Circle} - \text{Area of Minor Segment} \]

### Final Calculation
1. Calculate the area of the sector:
\[ \text{Area of Sector} = \frac{1}{6} \times \frac{22}{7} \times 196 = \frac{22 \times 196}{42} = \frac{4312}{42} \approx 102.67 \text{ cm}^2 \]
2. Calculate the area of the triangle:
\[ \text{Area of Triangle} = 49 \sqrt{3} \approx 49 \times 1.732 \approx 84.87 \text{ cm}^2 \]
3. Area of Minor Segment:
\[ \text{Area of Minor Segment} = 102.67 - 84.87 \approx 17.80 \text{ cm}^2 \]
4. Area of Circle:
\[ \text{Area of Circle} = \frac{22}{7} \times 196 \approx 615.43 \text{ cm}^2 \]
5. Area of Major Segment:
\[ \text{Area of Major Segment} = 615.43 - 17.80 \approx 597.63 \text{ cm}^2 \]

### Summary
- Area of Minor Segment: \( \approx 17.80 \text{ cm}^2 \)
- Area of Major Segment: \( \approx 597.63 \text{ cm}^2 \)