In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find :
Area of the segment formed by the corresponding chord.

Answer

To find the area of the segment formed by the chord in a circle of radius 21 cm, where the arc subtends an angle of 60° at the center, we will follow these steps:

### Step 1: Calculate the area of the sector
The area of the sector 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:
- \( r = 21 \) cm
- \( \theta = 60^\circ \)

Substituting the values:
\[ \text{Area of Sector} = \frac{60}{360} \times \pi \times (21)^2 \]
\[ = \frac{1}{6} \times \pi \times 441 \]
\[ = \frac{441\pi}{6} = 73.5\pi \text{ cm}^2 \]

### Step 2: Calculate the area of the triangle
The area of triangle OAB can be calculated using the formula:
\[ \text{Area of Triangle} = \frac{1}{2} \times AB \times OC \times \sin(\theta) \]
Here, \( AB \) is the length of the chord, which can be calculated using the formula:
\[ AB = 2r \sin\left(\frac{\theta}{2}\right) \]
Substituting the values:
\[ AB = 2 \times 21 \times \sin\left(\frac{60}{2}\right) = 42 \times \sin(30^\circ) = 42 \times \frac{1}{2} = 21 \text{ cm} \]

Now, substituting into the area of the triangle formula:
\[ \text{Area of Triangle} = \frac{1}{2} \times 21 \times 21 \times \sin(60^\circ) \]
\[ = \frac{1}{2} \times 21 \times 21 \times \frac{\sqrt{3}}{2} \]
\[ = \frac{441\sqrt{3}}{4} \text{ cm}^2 \]

### Step 3: Calculate the area of the segment
The area of the segment is given by:
\[ \text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle} \]
Substituting the values we calculated:
\[ \text{Area of Segment} = 73.5\pi - \frac{441\sqrt{3}}{4} \]

### Final Answer
Thus, the area of the segment formed by the chord is:
\[ \text{Area of Segment} = 73.5\pi - \frac{441\sqrt{3}}{4} \text{ cm}^2 \]