The image shows a sector of a circle with a central angle $\angle EFG = 120^\circ$, and the length of line segment $EF$ (which is the radius of the circle) is $24 \, \text{cm}$. You are asked to calculate the perimeter of the shaded area.

Step-by-step Solution:

1. Identify the elements of the sector:

• Radius $r = 24 \, \text{cm}$
• Central angle $\theta = 120^\circ$

2. Arc Length Calculation:

The formula for the length of an arc $L$ in a sector is: $L = 2\pi r \times \frac{\theta}{360^\circ}$ Substituting the known values: $L = 2\pi \times 24 \times \frac{120}{360} = 48\pi \times \frac{1}{3} = 16\pi \, \text{cm}$ Approximate this as: $L \approx 16 \times 3.14 = 50.24 \, \text{cm}$

3. Calculate the Perimeter of the Shaded Area:

The perimeter of the shaded area includes:

• The arc length ($L = 50.24 \, \text{cm}$),
• Two radii $EF = 24 \, \text{cm}$ and $FG = 24 \, \text{cm}$.

Thus, the total perimeter $P$ is: $P = L + EF + FG = 50.24 + 24 + 24 = 98.24 \, \text{cm}$

So, the perimeter of the shaded area is approximately 98.24 cm.

Do you want more details or have any questions?

Related Questions:

1. How would the perimeter change if the central angle was 90° instead of 120°?
2. Can you calculate the area of the shaded region?
3. What is the length of the circumference of the entire circle?
4. How do we calculate the area of a sector given a different radius?
5. How would the solution differ if the central angle was given in radians?

Tip:

Always ensure to convert angles into radians when working with formulas involving trigonometric functions.