Question

Area of sector of a circle with radius \(18 \text{ cm}\) is \(198 \text{ cm}^2\). The measure of central angle is

• A. \(70^{\circ}\)
• B. \(14^{\circ}\)
• C. \(140^{\circ}\)
• D. \(210^{\circ}\)

Hint:

Simplify numbers before multiplying everything out. For example, recognizing \(198 = 22 \times 9\) and \(18 \times 18 = 324\) makes the calculation much faster.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The area of a sector is a fraction of the total area of the circle based on the central angle \(\theta\).
Step 2: Key Formula or Approach:
Area of Sector = \(\frac{\theta}{360^{\circ}} \times \pi r^2\)
Step 3: Detailed Explanation:
Given: \(r = 18 \text{ cm}\), \(\text{Area} = 198 \text{ cm}^2\).
\[ 198 = \frac{\theta}{360^{\circ}} \times \frac{22}{7} \times 18 \times 18 \]
Simplify the equation:
\[ 198 = \frac{\theta}{360^{\circ}} \times \frac{22}{7} \times 324 \]
Divide 198 by 22:
\[ 9 = \frac{\theta}{360^{\circ}} \times \frac{1}{7} \times 324 \]
Divide 324 by 9:
\[ 1 = \frac{\theta}{360^{\circ}} \times \frac{1}{7} \times 36 \]
Now, simplify \(\frac{36}{360}\):
\[ 1 = \theta \times \frac{1}{10} \times \frac{1}{7} \]
\[ 1 = \frac{\theta}{70} \]
\[ \theta = 70^{\circ} \]
Step 4: Final Answer:
The measure of the central angle is \(70^{\circ}\).