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

Given:
Radius of circle \( r = 42 \) cm
Central angle of sector \( \theta = 30^\circ \)
Use \( \pi = \frac{22}{7} \)

Step 1: Area of minor sector
Formula:
\[ \text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]
Substitute values:
\[ = \frac{30}{360} \times \frac{22}{7} \times 42^2 \] Simplify:
\[ = \frac{1}{12} \times \frac{22}{7} \times 1764 \] \[ = \frac{22}{7} \times 147 \] \[ = 22 \times 21 = 462\ \text{cm}^2 \]
Area of the minor sector = 462 cm²

Step 2: Area of major sector
Major angle:
\[ 360^\circ - 30^\circ = 330^\circ \]
Area of major sector:
\[ \frac{330}{360} \times \pi r^2 \] \[ = \frac{11}{12} \times \frac{22}{7} \times 1764 \] \[ = \frac{22}{7} \times 1617 \] \[ = 22 \times 231 = 5082\ \text{cm}^2 \]
Area of the major sector = 5082 cm²

Final Answers:
• Area of minor sector = 462 cm²
• Area of major sector = 5082 cm²