Question

Area of a sector of a circle of radius 36 cm is \(54\pi\text{ cm}^2\). Find the central angle (in degrees) for corresponding arc of the sector.

• A. \(15^{\circ}\)
• B. \(45^{\circ}\)
• C. \(75^{\circ}\)
• D. \(105^{\circ}\)

Hint:

To simplify the algebra, write \((36)^2\) as \(36 \times 36\):
\[ 54 = \frac{\theta}{360} \times 36 \times 36 \]
Since \(\frac{36}{360} = \frac{1}{10}\), the equation becomes:
\[ 54 = \frac{36\theta}{10} \implies 5.4 = 0.36\theta \implies \theta = 15^{\circ} \]
This method reduces large division steps.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question is from the topic of Mensuration, specifically dealing with the area of a sector of a circle.
We are given the radius of the circle and the area of the sector. We need to find the measure of the central angle \(\theta\) in degrees.

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

Step 3: Detailed Explanation:
Given parameters:
Radius of the circle, \(r = 36\text{ cm}\)
Area of the sector \(= 54\pi\text{ cm}^2\)
Substitute these values into the sector area formula:
\[ 54\pi = \frac{\theta}{360^{\circ}} \times \pi \times (36)^2 \]
We can cancel \(\pi\) from both sides of the equation:
\[ 54 = \frac{\theta}{360^{\circ}} \times 1296 \]
Rearrange the terms to solve for \(\theta\):
\[ \theta = \frac{54 \times 360^{\circ}}{1296} \] Simplify the calculation:
First, divide 1296 by 36:
\[ \frac{1296}{36} = 36 \]
Rewrite the equation using this simplification:
\[ \theta = \frac{54 \times 10^{\circ}}{36} \]
Further simplify:
Divide 54 and 36 by 18:
\[ \frac{54}{18} = 3, \quad \frac{36}{18} = 2 \]
This gives:
\[ \theta = \frac{3 \times 10^{\circ}}{2} \]
\[ \theta = 3 \times 5^{\circ} = 15^{\circ} \]

Step 4: Final Answer:
The central angle of the sector is \(15^{\circ}\).