Question

The area of the sector of a circle of radius 42 cm and central angle 30\(^\circ\) is

• A. 515 cm\(^2\)
• B. 416 cm\(^2\)
• C. 462 cm\(^2\)
• D. 406 cm\(^2\)

Hint:

When performing calculations with fractions, always look for opportunities to cancel common factors before multiplying. This simplifies the arithmetic and reduces the chance of errors.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A sector of a circle is the portion of the circle enclosed by two radii and the arc between them. Its area is a fraction of the total area of the circle, determined by the central angle.

Step 2: Key Formula or Approach:
The area of a sector with central angle \(\theta\) (in degrees) and radius \(r\) is given by the formula:
\[ \text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]

Step 3: Detailed Explanation:
We are given:
Radius, \(r = 42\) cm
Central angle, \(\theta = 30^\circ\)
Substitute these values into the formula. We will use the approximation \(\pi = \frac{22}{7}\).
\[ \text{Area} = \frac{30}{360} \times \frac{22}{7} \times (42)^2 \] Simplify the fraction \(\frac{30}{360}\):
\[ \frac{30}{360} = \frac{3}{36} = \frac{1}{12} \] Now the calculation is:
\[ \text{Area} = \frac{1}{12} \times \frac{22}{7} \times 42 \times 42 \] We can simplify by canceling terms:
\[ \text{Area} = \frac{1}{12} \times 22 \times (6 \times 42) \] \[ \text{Area} = \frac{1}{2 \times 6} \times 22 \times 6 \times 42 \] Cancel the 6:
\[ \text{Area} = \frac{1}{2} \times 22 \times 42 \] \[ \text{Area} = 11 \times 42 \] \[ \text{Area} = 462 \text{ cm}^2 \]

Step 4: Final Answer:
The area of the sector is 462 cm\(^2\). This corresponds to option (C).