Question

Find the area of a sector whose angle is \( 30^\circ \) of a circle whose radius is 4 cm. Also, find the area of the corresponding major sector.

Hint:

When calculating the area of a sector, remember to use the angle in degrees and apply the formula for the sector area. For the major sector, subtract the minor sector area from the total circle area.

Answer 1

The area \( A \) of a sector of a circle is given by the formula: \[ A = \frac{\theta}{360^\circ} \times \pi r^2, \] where \( \theta \) is the angle of the sector, and \( r \) is the radius of the circle. For the given sector, \( \theta = 30^\circ \) and \( r = 4 \) cm. Substituting these values into the formula: \[ A = \frac{30^\circ}{360^\circ} \times \frac{22}{7} \times 4^2 = \frac{1}{12} \times \frac{22}{7} \times 16 = \frac{1}{12} \times \frac{352}{7} = \frac{352}{84} = \frac{88}{21} \approx 4.19 \, \text{cm}^2. \] Now, the area of the corresponding major sector is the total area of the circle minus the area of the minor sector. The total area of the circle is: \[ \text{Total area of the circle} = \pi r^2 = \frac{22}{7} \times 4^2 = \frac{22}{7} \times 16 = \frac{352}{7} \approx 50.29 \, \text{cm}^2. \] The area of the major sector is: \[ \text{Area of major sector} = \text{Total area} - \text{Area of minor sector} = 50.29 - 4.19 = 46.1 \, \text{cm}^2. \]
Conclusion:
- The area of the minor sector is approximately \( 4.19 \, \text{cm}^2 \).
- The area of the corresponding major sector is approximately \( 46.1 \, \text{cm}^2 \).