A chord of a circle of radius 12 cm subtends an angle of `120^@`at the centre. Find the area of the corresponding segment of the circle.

Explanation

To find the area of the segment of a circle that subtends an angle of \(120^\circ\) at the center, we will follow these steps:

### Step 1: Find 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 angle in degrees and \(r\) is the radius of the circle.

Given:
- \(\theta = 120^\circ\)
- \(r = 12 \, \text{cm}\)

Substituting the values into the formula:

\[ \text{Area of sector} = \frac{120}{360} \times \pi \times (12)^2 \]

Calculating this:

\[ = \frac{1}{3} \times \pi \times 144 \]
\[ = 48\pi \, \text{cm}^2 \]

### Step 2: Find the area of triangle OAB
To find the area of triangle OAB, we can use the formula:

\[ \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]

In triangle OAB, we need to find the base (AB) and the height (OB).

#### Step 2.1: Find the length of AB
Since the angle subtended at the center is \(120^\circ\), the angle at point O is \(60^\circ\) for triangle OAB (as it is half of \(120^\circ\)).

Using the sine function in triangle OAB:

\[ \sin(60^\circ) = \frac{AB}{OA} \]

Where \(OA = 12 \, \text{cm}\) (the radius).

Thus,

\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]
\[ \frac{\sqrt{3}}{2} = \frac{AB}{12} \]

Solving for AB:

\[ AB = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3} \, \text{cm} \]

#### Step 2.2: Find the height OB
Using the cosine function in triangle OAB:

\[ \cos(60^\circ) = \frac{OB}{OA} \]

Thus,

\[ \cos(60^\circ) = \frac{1}{2} \]
\[ \frac{1}{2} = \frac{OB}{12} \]

Solving for OB:

\[ OB = 12 \times \frac{1}{2} = 6 \, \text{cm} \]

#### Step 2.3: Calculate the area of triangle OAB
Now we can find the area of triangle OAB:

\[ \text{Area of triangle OAB} = \frac{1}{2} \times AB \times OB \]
\[ = \frac{1}{2} \times (6\sqrt{3}) \times 6 \]
\[ = 18\sqrt{3} \, \text{cm}^2 \]

### Step 3: Find the area of the segment
The area of the segment is given by:

\[ \text{Area of segment} = \text{Area of sector} - \text{Area of triangle OAB} \]

Substituting the values we calculated:

\[ \text{Area of segment} = 48\pi - 18\sqrt{3} \, \text{cm}^2 \]

### Step 4: Approximate the area of the segment
Using \(\pi \approx 3.14\) and \(\sqrt{3} \approx 1.73\):

\[ 48\pi \approx 48 \times 3.14 = 150.72 \, \text{cm}^2 \]
\[ 18\sqrt{3} \approx 18 \times 1.73 = 31.14 \, \text{cm}^2 \]

Thus,

\[ \text{Area of segment} \approx 150.72 - 31.14 = 119.58 \, \text{cm}^2 \]

### Final Answer
The area of the corresponding segment of the circle is approximately \(119.58 \, \text{cm}^2\).

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