Question

The area (in \(\text{cm}^2\)) of a sector of a circle of radius 14 cm cut off by an arc of length 22 cm is :

• A. 77
• B. 308
• C. 154
• D. 462

Hint:

Using the formula \(A = \frac{1}{2} \times l \times r\) is highly efficient.
It bypasses the need to find the central angle \(\theta\), which would require multiple steps of algebraic manipulation.
Think of it like the area of a triangle formula \(A = \frac{1}{2} \times \text{base} \times \text{height}\), where the arc length acts as the base and the radius acts as the height!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
A sector is cut off from a circle of radius 14 cm. The length of the boundary arc of this sector is 22 cm. We need to calculate the area of this sector.

Step 2: Key Formula or Approach:
1. The standard formulas for arc length (\(l\)) and sector area (\(A\)) for a sector with central angle \(\theta\) (in degrees) and radius \(r\) are:
\[ l = \frac{\theta}{360^\circ} \times 2\pi r \]
\[ A = \frac{\theta}{360^\circ} \times \pi r^2 \]
2. By dividing the two formulas, we can derive a direct relationship between the area of the sector, its arc length, and its radius:
\[ A = \frac{1}{2} \times l \times r \]

Step 3: Detailed Explanation:
1. Identify the given parameters:
Radius of the circle, \(r = 14\ \text{cm}\)
Length of the arc, \(l = 22\ \text{cm}\)
2. Use the direct formula relating the area of a sector to the arc length and radius:
\[ \text{Area of Sector } (A) = \frac{1}{2} \times l \times r \]
3. Substitute the values of \(l\) and \(r\):
\[ A = \frac{1}{2} \times 22 \times 14 \]
4. Simplify the calculation:
\[ A = 11 \times 14 \]
\[ A = 154\ \text{cm}^2 \]
This gives the area of the sector as \(154\ \text{cm}^2\).

Step 4: Final Answer:
The area of the sector is \(154\ \text{cm}^2\), which corresponds to option (C).