In a circle of radius 21 cm, an arc subtends an angle of `60^o`at the centre. Find:(i) the length of the arc (ii) area of the sector formed by the arc (iii) area of the segment formed by the corresponding chord

Explanation

To solve the problem step by step, we will find the length of the arc, the area of the sector, and the area of the segment formed by the corresponding chord in a circle of radius 21 cm with an angle of 60° at the center.

### Step 1: Find the Length of the Arc

The formula for the length of an arc (L) is given by:

\[ L = \frac{\theta}{360} \times 2 \pi r \]

where:
- \(\theta\) is the angle subtended at the center (in degrees),
- \(r\) is the radius of the circle.

Given:
- \(\theta = 60^\circ\)
- \(r = 21 \, \text{cm}\)

Substituting the values into the formula:

\[ L = \frac{60}{360} \times 2 \times \frac{22}{7} \times 21 \]

Now, simplify the expression:

\[ L = \frac{1}{6} \times 2 \times \frac{22}{7} \times 21 \]

Calculating step by step:

1. \(\frac{60}{360} = \frac{1}{6}\)
2. \(2 \times \frac{22}{7} = \frac{44}{7}\)
3. Now, multiply \(\frac{44}{7} \times 21\):

\[ \frac{44 \times 21}{7} = \frac{924}{7} = 132 \]

4. Finally, multiply by \(\frac{1}{6}\):

\[ L = \frac{132}{6} = 22 \, \text{cm} \]

**Length of the arc = 22 cm.**

### Step 2: Find the Area of the Sector

The formula for the area of a sector (A) is given by:

\[ A = \frac{\theta}{360} \times \pi r^2 \]

Substituting the values:

\[ A = \frac{60}{360} \times \pi \times (21)^2 \]

Simplifying:

1. \(\frac{60}{360} = \frac{1}{6}\)
2. \(\pi \times (21)^2 = \pi \times 441 = \frac{22}{7} \times 441\)

Now, multiply:

\[ A = \frac{1}{6} \times \frac{22}{7} \times 441 \]

Calculating step by step:

1. \(\frac{441}{6} = 73.5\)
2. Now multiply by \(\frac{22}{7}\):

\[ A = \frac{22 \times 73.5}{7} = \frac{1617}{7} = 231 \, \text{cm}^2 \]

**Area of the sector = 231 cm².**

### Step 3: Find the Area of the Segment

The area of the segment is given by:

\[ \text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle} \]

We already calculated the area of the sector as 231 cm². Now we need to find the area of triangle \(AOB\).

The formula for the area of a triangle when two sides and the included angle are known is:

\[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(\theta) \]

Where:
- \(a = 21 \, \text{cm}\)
- \(b = 21 \, \text{cm}\)
- \(\theta = 60^\circ\)

Substituting the values:

\[ \text{Area of Triangle} = \frac{1}{2} \times 21 \times 21 \times \sin(60^\circ) \]

Calculating step by step:

1. \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)
2. Area = \(\frac{1}{2} \times 21 \times 21 \times \frac{\sqrt{3}}{2} = \frac{441\sqrt{3}}{4}\)

Now substituting back to find the area of the segment:

\[ \text{Area of Segment} = 231 - \frac{441\sqrt{3}}{4} \]

This is the area of the segment formed by the chord.

**Area of the segment = \(231 - \frac{441\sqrt{3}}{4} \, \text{cm}^2\).**