An arc of a circle of radius 14cm subtends angle 300° at the centre. Find the perimeter of the sector formed by the arc (take π = $\frac{22}{7}$)

Analysis

To find the perimeter of a sector, we need to calculate the sum of the length of the arc and the two radii that enclose the sector.

Concept

1. Arc Length Formula: The length of an arc ($L$) is given by: $L = \frac{\theta}{360} \times 2\pi r$ where $\theta$ is the angle subtended at the center and $r$ is the radius.
2. Perimeter of a Sector: The perimeter ($P$) is the sum of the arc length and the two radii: $P = L + 2r$

Step-by-Step Solution

Step 1: Identify the given values

• Radius ($r$) = 14 cm
• Angle ($\theta$) = 300°
• $\pi = \frac{22}{7}$

Step 2: Calculate the Arc Length ($L$) $L = \frac{300}{360} \times 2 \times \frac{22}{7} \times 14$

Simplify the fraction $\frac{300}{360}$: $\frac{300}{360} = \frac{30}{36} = \frac{5}{6}$

Now, calculate $L$: $L = \frac{5}{6} \times 2 \times \frac{22}{7} \times 14$ $L = \frac{5}{6} \times 2 \times 22 \times 2$ (since $14 \div 7 = 2$) $L = \frac{5 \times 88}{6} = \frac{440}{6}$ $L \approx 73.33 \text{ cm}$

Step 3: Calculate the Perimeter ($P$) The perimeter of the sector is the arc length plus the two radii forming the sector: $P = L + r + r$ $P = 73.33 + 14 + 14$ $P = 73.33 + 28$ $P = 101.33 \text{ cm}$

Conclusion

Comparing our result with the given options, we find that 101.33 cm matches option C.

Exam Tip

Always remember that a "sector" is like a slice of pizza. The perimeter includes not just the curved crust (the arc), but also the two straight sides (the radii). Students often forget to add the two radii and mistakenly choose the arc length alone!