Question

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

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

Hint:

Before multiplying out large numbers, always check if they can be cancelled with the opposite side of the equation.
In this problem, the product \(22 \times 9 = 198\) cancels out perfectly with the \(198\) on the left side, saving you from doing long division!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The topic of this question is Areas Related to Circles.
A sector is a portion of a circle enclosed by two radii and an arc.
The area of a sector depends on the radius of the circle and the measure of the central angle \(\theta\) subtended by the arc at the center.
We are given a radius \(r = 18\) cm and the area of the sector as \(198 \text{ cm}^2\). We need to calculate the central angle \(\theta\).

Step 2: Key Formula or Approach:
The area of a sector of a circle with radius \(r\) and central angle \(\theta\) is given by the formula:
\[ \text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 \] We will substitute the given values of the Area and \(r\) into this formula, use \(\pi = \frac{22}{7}\), and solve for \(\theta\).

Step 3: Detailed Explanation:

• Write down the formula and substitute the known values:
\[ 198 = \frac{\theta}{360^\circ} \times \frac{22}{7} \times 18 \times 18 \]

• Simplify the expression step-by-step:
Divide 360 by 18:
\[ \frac{18 \times 18}{360} = \frac{18}{20} = \frac{9}{10} \] Now substitute this back:
\[ 198 = \theta \times \frac{22}{7} \times \frac{9}{10} \]

• Solve for \(\theta\):
Multiply the terms:
\[ 22 \times 9 = 198 \] So the equation becomes:
\[ 198 = \theta \times \frac{198}{70} \]

• Cancel out the common factor of 198 from both sides:
\[ 1 = \frac{\theta}{70^\circ} \] \[ \theta = 70^\circ \]

Step 4: Final Answer:
The measure of the central angle is \(70^\circ\).
Therefore, the correct option is (A).