https://file.aiquickdraw.com/mathbot/file/1726283468032l77y62f2.jpg

The problem involves a sector of a circle with center \( O \), radius \( 7 \, \text{cm} \), and an angle \( \angle AOB = 30^\circ \).

### Part (a): Find the perimeter of the sector

The perimeter of the sector is the sum of the two radii \( OA \) and \( OB \), plus the arc length \( AB \). Here's how you can calculate it:

1. **Arc Length Calculation**:
   The formula for the arc length \( l \) of a sector is:

   \[
   l = \theta \times r
   \]
   where \( \theta \) is the angle in radians, and \( r \) is the radius of the circle.

   First, we convert the angle \( 30^\circ \) to radians:

   \[
   \theta = \frac{30^\circ}{360^\circ} \times 2\pi = \frac{\pi}{6} \, \text{radians}
   \]

   Now, using \( \pi = \frac{22}{7} \), we calculate:

   \[
   l = \frac{\pi}{6} \times 7 = \frac{22}{7} \times \frac{7}{6} = \frac{22}{6} = 3.67 \, \text{cm}
   \]

2. **Perimeter of the Sector**:
   The total perimeter is the sum of the arc length and the two radii:

   \[
   \text{Perimeter} = OA + OB + AB = 7 + 7 + 3.67 = 17.67 \, \text{cm}
   \]

So, the perimeter of the sector is approximately \( 17.67 \, \text{cm} \).

Would you like details on part (b) or any further clarifications?

### Follow-up Questions:
1. What is the formula for the area of a sector?
2. How does the perimeter of a sector change if the central angle increases?
3. How would the arc length change if the radius were doubled but the angle remained the same?
4. Can you calculate the area of this sector using the given information?
5. What is the relationship between the arc length and the circumference of the circle?

### Tip:
Always ensure that angles are converted to radians when using them in formulas involving trigonometry or arc lengths, unless working with degrees is explicitly required.

The diagram shows a sector OAB of a circle, center O, radius 7 cm, in which angle AOB = 30°. Taking π to be 22/7, calculate the perimeter of the sector.

This problem involves calculating the perimeter of a sector with a radius of 7 cm and an angle of 30°. Using the arc length formula and geometric principles of circles, we find that the perimeter is approximately 17.67 cm. Suitable for Grades 8-10 students.

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