A sector has a radius of 12 cm and a central angle of 65°. Determine a. the measure of the central angle in radians,
b. the area of the sector, and
c. the perimeter of the sector

To solve the problem, we'll follow these steps:

### Given:
- Radius \(r = 12\) cm
- Central angle \(\theta = 65^\circ\)

### (a) Convert the central angle to radians:
To convert the central angle from degrees to radians, use the formula:
\[
\text{Radians} = \text{Degrees} \times \frac{\pi}{180}
\]
\[
\text{Radians} = 65^\circ \times \frac{\pi}{180} = \frac{65\pi}{180} = \frac{13\pi}{36} \text{ Radians} \approx 1.134 \text{ Radians}
\]

### (b) Calculate the area of the sector:
The area \(A\) of a sector is given by the formula:
\[
A = \frac{1}{2} r^2 \theta
\]
where \(r\) is the radius and \(\theta\) is the central angle in radians.

Substituting the values:
\[
A = \frac{1}{2} \times 12^2 \times \frac{13\pi}{36} = \frac{1}{2} \times 144 \times \frac{13\pi}{36} = 72 \times \frac{13\pi}{36} = 26\pi \text{ cm}^2 \approx 81.68 \text{ cm}^2
\]

### (c) Calculate the perimeter of the sector:
The perimeter \(P\) of the sector is the sum of the arc length and the two radii. The arc length \(L\) is given by:
\[
L = r \theta
\]

So, the perimeter is:
\[
P = 2r + L = 2r + r\theta = 2 \times 12 + 12 \times \frac{13\pi}{36} = 24 + 12 \times \frac{13\pi}{36} 
\]
\[
P = 24 + \frac{13\pi}{3} \text{ cm} \approx 37.63 \text{ cm}
\]

### Final Answers:
1. **Measure of the central angle in radians:** \(\frac{13\pi}{36} \approx 1.134\) radians.
2. **Area of the sector:** \(26\pi \text{ cm}^2 \approx 81.68 \text{ cm}^2\)
3. **Perimeter of the sector:** \(24 + \frac{13\pi}{3} \text{ cm} \approx 37.63 \text{ cm}\)

Would you like more details or have any questions?

### Follow-up Questions:
1. How do you derive the formula for the area of a sector?
2. What is the arc length for a central angle of \(90^\circ\) and radius \(10\) cm?
3. How do you convert a central angle in radians back to degrees?
4. What would be the perimeter if the central angle was \(90^\circ\)?
5. How does the area of a sector change if the radius is doubled?
6. Can you find the area of a sector with a central angle of \(1\) radian and radius \(7\) cm?
7. What is the formula for the circumference of a full circle?
8. How would you calculate the area of a segment, which is a part of a sector?

#### Tip:
Always ensure that when calculating the area of a sector, the angle is in radians.

Learn how to calculate the central angle in radians, area, and perimeter of a sector with a radius of 12 cm and a central angle of 65°. Includes step-by-step solutions and formulas for geometry and trigonometry. Suitable for Grades 9-12 students.

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