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: Use the sector area formula in radians instead of degrees.
For a sector of radius $r$ and angle $\theta$ measured in radians, \[ \text{Area} = \frac{1}{2} r^2 \theta \] Here $r = 36$ cm and Area $= 54\pi$ cm$^2$.
Step 2: Solve for theta in radians. \[ 54\pi = \frac{1}{2}(36)^2 \theta \] \[ 54\pi = 648\,\theta \] \[ \theta = \frac{54\pi}{648} = \frac{\pi}{12} \]
Step 3: Convert this angle to degrees.
Since $\pi$ radians is $180^{\circ}$, \[ \theta = \frac{\pi}{12} \times \frac{180^{\circ}}{\pi} = 15^{\circ} \]
Step 4: Conclude. \[ \boxed{15^{\circ}} \]