If the angle of a sector of a circle with radius 10.5 cm is 120°, find the perimeter of the sector. [Take $\pi = \frac{22}{7}$]

Analysis

To find the perimeter of a sector, we must remember that a sector is bounded by two radii and an arc. Therefore, the perimeter is the sum of the length of the arc and the two radii.

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 of the sector and $r$ is the radius.
2. Perimeter Formula: The perimeter ($P$) of a sector is: $P = L + 2r$

Step-by-Step Solution

Step 1: Calculate the Arc Length ($L$) Given: $\theta = 120^\circ$, $r = 10.5\text{ cm}$, $\pi = \frac{22}{7}$ $L = \frac{120}{360} \times 2 \times \frac{22}{7} \times 10.5$

Simplify the fraction: $\frac{120}{360} = \frac{1}{3}$

Now calculate: $L = \frac{1}{3} \times 2 \times \frac{22}{7} \times 10.5$ $L = \frac{1}{3} \times \frac{44}{7} \times 10.5$ $L = \frac{44 \times 10.5}{21}$ $L = \frac{462}{21} = 22\text{ cm}$

Step 2: Calculate the Perimeter ($P$) $P = L + 2r$ $P = 22 + 2(10.5)$ $P = 22 + 21$ $P = 43\text{ cm}$

Conclusion

The perimeter of the sector is 43 cm. This matches option C.

Exam Tip

Always remember that a sector is like a slice of pizza. When asked for the "perimeter," students often calculate only the arc length (the crust) and forget to add the two straight sides (the radii). Don't fall for that trap! Always add $2r$ to your arc length.