Question

The area (in sq cm) of a sector whose radius is 18 cm and angle measure of \(30^\circ\) is :

• A. \(3 \pi\)
• B. \(18 \pi\)
• C. \(27 \pi\)
• D. \(54 \pi\)

Hint:

Formula for Area of a Sector: \(A = \frac{\theta}{360^\circ} \times \pi r^2\). 1. Given: radius \(r = 18\) cm, angle \(\theta = 30^\circ\). 2. Calculate the fraction of the circle: \(\frac{30}{360} = \frac{1}{12}\). 3. Calculate \(r^2\): \(18^2 = 324\). 4. Area = \(\frac{1}{12} \times \pi \times 324\). 5. Simplify: \(\frac{324}{12} = 27\). 6. Result: Area = \(27\pi\) sq cm.

Answer 1

The Correct Option is C

Concept: The area of a sector of a circle with radius \(r\) and central angle \(\theta\) (in degrees) is given by the formula: Area of sector = \(\frac{\theta}{360^\circ} \times \pi r^2\). Step 1: Identify the given values
Radius of the sector, \(r = 18 \text{ cm}\).
Angle of the sector, \(\theta = 30^\circ\). Step 2: Substitute the values into the formula for the area of a sector \[ \text{Area} = \frac{30^\circ}{360^\circ} \times \pi (18)^2 \] Step 3: Simplify the fraction \(\frac{\theta}{360^\circ}\) \[ \frac{30}{360} = \frac{3}{36} = \frac{1}{12} \] Step 4: Calculate \(r^2\) \[ (18)^2 = 18 \times 18 = 324 \] Step 5: Calculate the area of the sector \[ \text{Area} = \frac{1}{12} \times \pi \times 324 \] \[ \text{Area} = \frac{324\pi}{12} \] To simplify \(\frac{324}{12}\): Divide by common factors. Both are divisible by 2: \(\frac{162}{6}\). Divide by 2 again: \(\frac{81}{3}\). Divide by 3: \(27\). So, \(\frac{324}{12} = 27\). \[ \text{Area} = 27\pi \text{ sq cm} \] The area of the sector is \(27\pi \text{ cm}^2\). This matches option (3).