Question

The perimeter of sector of a circle of radius 21 cm and central angle 60\(^\circ\), is

• A. 22 cm
• B. 44 cm
• C. 64 cm
• D. 273 cm

Hint:

A common mistake is calculating only the arc length (22 cm) and choosing Option (A).
Remember, a sector is bounded by two radii as well, so always add \(2r\) to the arc length!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the perimeter of a sector of a circle of radius \(r = 21 \text{ cm}\) with a central angle of \(\theta = 60^\circ\).

Step 2: Key Formula or Approach:
- The perimeter of a sector is the sum of the arc length and the lengths of the two bounding radii:
\[ \text{Perimeter} = \text{Arc Length } (l) + 2r \]
- The formula for the length of an arc is:
\[ l = \frac{\theta}{360} \times 2\pi r \]

Step 3: Detailed Explanation:

• Find the length of the arc \(l\):
Substitute \(\theta = 60^\circ\), \(r = 21 \text{ cm}\), and \(\pi = \frac{22}{7}\):
\[ l = \frac{60}{360} \times 2 \times \frac{22}{7} \times 21 \]
Simplify the terms:
\[ l = \frac{1}{6} \times 2 \times 22 \times 3 \]
\[ l = \frac{1}{6} \times 132 = 22 \text{ cm} \]

• Calculate the total perimeter of the sector:
\[ \text{Perimeter} = l + 2r \]
\[ \text{Perimeter} = 22 + 2(21) \]
\[ \text{Perimeter} = 22 + 42 = 64 \text{ cm} \]


Step 4: Final Answer:
The perimeter of the sector is 64 cm.