Unless stated otherwise, use the approximation for . 4. Find the perimeter of a sector (i.e., the curved portion as well as the two straight portions) of a circle of radius and sector angle .

Understand the Question The perimeter of a sector is the total boundary length enclosing the sector, which consists of the curved arc length () and the two straight radii (): Given: Radius, Sector angle, First calculate the arc length , then add to get the total perimeter. Step 1 · Find the Arc Length Arc length of a sector is given by Substitute , , and Step 2 · Calculate the Perimeter of the Sector The perimeter includes the arc length and the two bounding radii Substitute and Answer > (or ) Common Mistakes Forgetting the Radii: Calculating only the arc length and omitting the two straight radii (). The perimeter of a sector is always . Adding Only One Radius: Adding instead of . A sector is enclosed by two radii meeting at the center.

Measuring Space: Perimeter and Area | Exercise 6.1

Question 4

Unless stated otherwise, use the approximation $\dfrac{22}{7}$ for $\pi$ .

Find the perimeter of a sector (i.e., the curved portion as well as the two straight portions) of a circle of radius $14\text{ cm}$ and sector angle $75^\circ$ .

Solution

Understand the Question

The perimeter of a sector is the total boundary length enclosing the sector, which consists of the curved arc length ( $L$ ) and the two straight radii ( $2r$ ): $\text{Perimeter} = L + 2r$
Given:
- Radius, $r = 14\text{ cm}$
- Sector angle, $\theta = 75^\circ$
- $\pi = \dfrac{22}{7}$
First calculate the arc length $L = \dfrac{\theta}{360^\circ} \times 2\pi r$ , then add $2r$ to get the total perimeter.

Step 1 · Find the Arc Length

Diagram 1

Arc length of a sector is given by $L = \dfrac{\theta}{360^\circ} \times 2\pi r$

Substitute $r = 14\text{ cm}$ , $\theta = 75^\circ$ , and $\pi = \dfrac{22}{7}$

\begin{aligned} L &= \dfrac{75}{360} \times 2 \times \dfrac{22}{7} \times 14 \\[0.6em] &= \dfrac{75}{360} \times 2 \times 22 \times 2 \\[0.6em] &= \dfrac{75}{360} \times 88 \\[0.6em] &= \dfrac{5}{24} \times 88 \\[0.6em] &= \dfrac{5 \times 11}{3} \\[0.6em] &= \dfrac{55}{3}\text{ cm} \end{aligned}

Step 2 · Calculate the Perimeter of the Sector

The perimeter includes the arc length and the two bounding radii $\text{Perimeter} = L + 2r$

Substitute $L = \dfrac{55}{3}\text{ cm}$ and $r = 14\text{ cm}$

\begin{aligned} \text{Perimeter} &= \dfrac{55}{3} + 2 \times 14 \\[0.6em] &= \dfrac{55}{3} + 28 \\[0.6em] &= \dfrac{55}{3} + \dfrac{84}{3} \\[0.6em] &= \dfrac{139}{3} \text{ cm} \\[0.6em] &= 46.33 \text{ cm (approx.)} \end{aligned}

Answer

$\dfrac{139}{3}\text{ cm}$ (or $\approx 46.33\text{ cm}$ )

Common Mistakes

Forgetting the Radii: Calculating only the arc length $L$ and omitting the two straight radii ( $2r$ ). The perimeter of a sector is always $L + 2r$ .
Adding Only One Radius: Adding $L + r$ instead of $L + 2r$ . A sector is enclosed by two radii meeting at the center.

Question 4

More questions in Exercise 6.1