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 \(\pi = \frac{22}{7}\)]

• A. Minor Area = 462 \(\text{cm}^2\), Major Area = 5082 \(\text{cm}^2\)
• B. Minor Area = 462 \(\text{cm}^2\), Major Area = 5544 \(\text{cm}^2\)
• C. Minor Area = 231 \(\text{cm}^2\), Major Area = 5082 \(\text{cm}^2\)
• D. Minor Area = 154 \(\text{cm}^2\), Major Area = 5082 \(\text{cm}^2\)

Hint:

Using \(A_{\text{major}} = \text{Total Area} - A_{\text{minor}}\) is much faster and less prone to calculation errors than calculating with large fraction angles like \(\frac{330^\circ}{360^\circ}\).
Always calculate the total area first if both sectors are required!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a circle with a radius \(r = 42\ \text{cm}\) and a central angle \(\theta = 30^\circ\). We need to calculate:
1. The area of the minor sector.
2. The area of the corresponding major sector.

Step 2: Key Formula or Approach:
1. The area of a minor sector with a central angle \(\theta\) is:
\[ A_{\text{minor}} = \frac{\theta}{360^\circ} \times \pi r^2 \]
2. The area of the major sector can be calculated by:
- Subtracting the minor sector area from the total area of the circle:
\[ A_{\text{major}} = \pi r^2 - A_{\text{minor}} \]
- Or using the remaining angle \((360^\circ - \theta)\):
\[ A_{\text{major}} = \frac{360^\circ - \theta}{360^\circ} \times \pi r^2 \]

Step 3: Detailed Explanation:
1.

Calculate the area of the minor sector:
Given \(r = 42\ \text{cm}\), \(\theta = 30^\circ\), and \(\pi = \frac{22}{7}\).
\[ A_{\text{sector}} = \frac{30^\circ}{360^\circ} \times \frac{22}{7} \times 42 \times 42 \]
Simplify the fraction:
\[ \frac{30^\circ}{360^\circ} = \frac{1}{12} \]
Substitute and calculate:
\[ A_{\text{sector}} = \frac{1}{12} \times \frac{22}{7} \times 14 \times 3 \times 14 \]
Alternatively, group the numbers:
\[ A_{\text{sector}} = \frac{1}{12} \times 22 \times 6 \times 42 \]
Since \(12 = 6 \times 2\):
\[ A_{\text{sector}} = \frac{22 \times 42}{2} = 11 \times 42 = 462\ \text{cm}^2 \]
The area of the minor sector is \(462\ \text{cm}^2\).

2.

Calculate the area of the major sector:
The central angle of the major sector is:
\[ 360^\circ - 30^\circ = 330^\circ \]
Using the major sector formula:
\[ A_{\text{major}} = \frac{330^\circ}{360^\circ} \times \frac{22}{7} \times 42 \times 42 \]
Simplify the fraction:
\[ \frac{330^\circ}{360^\circ} = \frac{11}{12} \]
Substitute and calculate:
\[ A_{\text{major}} = \frac{11}{12} \times 22 \times 6 \times 42 \]
\[ A_{\text{major}} = \frac{11 \times 22 \times 42}{2} = 11 \times 11 \times 42 \]
\[ A_{\text{major}} = 121 \times 42 = 5082\ \text{cm}^2 \]
(Alternatively: \(\text{Total Area} = \pi r^2 = \frac{22}{7} \times 42 \times 42 = 22 \times 6 \times 42 = 5544\ \text{cm}^2\).
\(A_{\text{major}} = 5544 - 462 = 5082\ \text{cm}^2\)).

Step 4: Final Answer:
The area of the minor sector is \(462\ \text{cm}^2\) and the major sector is \(5082\ \text{cm}^2\), which corresponds to option (A).