Question

Find the area of the sector of a circle of radius 42 cm and of central angle \( 30^\circ \). Also, find the area of the corresponding major sector. [Use \( \pi = \frac{22}{7} \)]

Hint:

Using \( 360^\circ - \theta \) for the major sector is often faster than calculating the full circle area and subtracting.

Answer 1

Step 1: Understanding the Concept:
The area of a sector depends on the central angle and the total area of the circle.
Step 2: Key Formula or Approach:
Area of sector \( = \frac{\theta}{360} \times \pi r^2 \).
Area of major sector \( = \text{Total Area} - \text{Area of minor sector} \).
Step 3: Detailed Explanation:
Given: \( r = 42 \text{ cm} \), \( \theta = 30^\circ \).
1. Area of minor sector:
\[ \text{Area} = \frac{30}{360} \times \frac{22}{7} \times 42 \times 42 \]
\[ \text{Area} = \frac{1}{12} \times 22 \times 6 \times 42 \]
\[ \text{Area} = \frac{1}{2} \times 22 \times 42 = 11 \times 42 = 462 \text{ cm}^2 \]
2. Area of major sector:
The central angle of the major sector is \( 360^\circ - 30^\circ = 330^\circ \).
\[ \text{Area of major sector} = \frac{330}{360} \times \frac{22}{7} \times 42 \times 42 \]
\[ \text{Area of major sector} = \frac{11}{12} \times 22 \times 6 \times 42 \]
\[ \text{Area of major sector} = 11 \times 11 \times 42 = 121 \times 42 = 5082 \text{ cm}^2 \]
Alternatively: Total Area \( = \frac{22}{7} \times 42 \times 42 = 5544 \text{ cm}^2 \).
Major sector area \( = 5544 - 462 = 5082 \text{ cm}^2 \).
Step 4: Final Answer:
Minor Sector Area \( = 462 \text{ cm}^2 \); Major Sector Area \( = 5082 \text{ cm}^2 \).