Question

The area of a sector of a circle is \(110\text{ cm}^2\) and the central angle of the sector is \(56^\circ\). Find the radius of the circle. Take \(\pi = \dfrac{22}{7}\).

• A. \(35\text{ cm}\)
• B. \(20\text{ cm}\)
• C. \(25\text{ cm}\)
• D. \(15\text{ cm}\)

Hint:

For sector problems: \[ \text{Sector Area}= \frac{\theta}{360}\pi r^2 \] Simplify fractions early to avoid large calculations.

Answer 1

The Correct Option is D

Concept: The area of a sector is given by: \[ \text{Area of Sector} = \frac{\theta}{360}\pi r^2 \] where:

• \(\theta\) = central angle
• \(r\) = radius

Step 1: Substitute the given values. Given: \[ \text{Area} = 110\text{ cm}^2 \] \[ \theta = 56^\circ \] \[ \pi = \frac{22}{7} \] Thus, \[ 110 = \frac{56}{360}\times\frac{22}{7}\times r^2 \]

Step 2: Simplify the fraction. \[ 110 = \frac{7}{45}\times\frac{22}{7}\times r^2 \] \[ 110 = \frac{22}{45}r^2 \]

Step 3: Solve for \(r^2\). \[ r^2 = 110\times\frac{45}{22} \] \[ = 5\times45 \] \[ = 225 \] \[ r = \sqrt{225} \] \[ r = 15 \] Hence, \[ \boxed{15\text{ cm}} \]