Question 4

We need to find the total length around the sector.

Step 1 — Find Arc Length

First, let's find the length of the curved part. This is called the arc length.

$L = \frac{\theta}{360^\circ} \times 2\pi r$

We are given the radius $r = \textbf{14 cm}$. The sector angle $\theta = \textbf{75°}$. Let's use $\pi = \frac{22}{7}$.

$L = \frac{75}{360} \times 2 \times \frac{22}{7} \times 14$

$= \frac{75}{360} \times 2 \times 22 \times 2$

$= \frac{75}{360} \times 88$

$= \frac{5}{24} \times 88$

$= \frac{5 \times 11}{3}$

$\boxed{\frac{55}{3} \text{ cm}}$

Step 2 — Calculate Perimeter

The perimeter of the sector includes the arc length and two straight radii. Let's add these lengths together.

$\text{Perimeter} = L + 2r$

We found the arc length $L = \frac{55}{3} \text{ cm}$. The radius $r = \textbf{14 cm}$.

$\text{Perimeter} = \frac{55}{3} + 2 \times 14$

$= \frac{55}{3} + 28$

$= \frac{55}{3} + \frac{84}{3}$

$= \frac{139}{3} \text{ cm}$

We can also write this as a decimal.

$= 46.33 \text{ cm (approx.)}$

Answer

The perimeter of the sector is $\frac{139}{3} \text{ cm}$ or $46.33 \text{ cm}$.