Find the area of both the segments of a circle of raidus 42 cm with central angle `120^(@)`. [Given `sin 120^(@)=(sqrt(3))/2` and `sqrt(3)=1.73`]

Explanation

To find the area of both segments of a circle with a radius of 42 cm and a central angle of 120°, we can follow these steps:

### Step 1: Calculate the area of the entire circle.
The formula for the area of a circle is given by:
\[ \text{Area} = \pi r^2 \]
Where \( r \) is the radius.

Given:
- \( r = 42 \, \text{cm} \)
- \( \pi \approx \frac{22}{7} \)

Calculating the area:
\[ \text{Area} = \frac{22}{7} \times (42)^2 \]
\[ = \frac{22}{7} \times 1764 \]
\[ = \frac{38708}{7} \approx 5530.57 \, \text{cm}^2 \]

### Step 2: Calculate the area of the sector.
The area of a sector can be calculated using the formula:
\[ \text{Area of Sector} = \frac{\theta}{360} \times \text{Area of Circle} \]
Where \( \theta \) is the central angle in degrees.

Given:
- \( \theta = 120° \)

Calculating the area of the sector:
\[ \text{Area of Sector} = \frac{120}{360} \times 5530.57 \]
\[ = \frac{1}{3} \times 5530.57 \approx 1843.52 \, \text{cm}^2 \]

### Step 3: Calculate the area of the triangle formed by the radii and the chord.
The area of the triangle can be calculated using:
\[ \text{Area of Triangle} = \frac{1}{2} \times a \times b \times \sin(\theta) \]
Where \( a \) and \( b \) are the lengths of the sides (both equal to the radius), and \( \theta \) is the angle between them.

Given:
- \( a = b = 42 \, \text{cm} \)
- \( \sin(120°) = \frac{\sqrt{3}}{2} \approx 0.866 \)

Calculating the area of the triangle:
\[ \text{Area of Triangle} = \frac{1}{2} \times 42 \times 42 \times \frac{\sqrt{3}}{2} \]
\[ = \frac{1}{2} \times 1764 \times 0.866 \approx 762.93 \, \text{cm}^2 \]

### Step 4: Calculate the area of the smaller segment.
The area of the smaller segment is given by:
\[ \text{Area of Smaller Segment} = \text{Area of Sector} - \text{Area of Triangle} \]
Calculating:
\[ \text{Area of Smaller Segment} = 1843.52 - 762.93 \approx 1080.59 \, \text{cm}^2 \]

### Step 5: Calculate the area of the larger segment.
The area of the larger segment can be found by subtracting the area of the smaller segment from the area of the sector:
\[ \text{Area of Larger Segment} = \text{Area of Circle} - \text{Area of Sector} - \text{Area of Triangle} \]
Calculating:
\[ \text{Area of Larger Segment} = 5530.57 - 1843.52 - 762.93 \approx 3924.12 \, \text{cm}^2 \]

### Final Results:
- Area of the smaller segment: \( 1080.59 \, \text{cm}^2 \)
- Area of the larger segment: \( 3924.12 \, \text{cm}^2 \)