The perimeter of a certain sector of a circle is equal to the length of the arc of the semicircle having the same radius. The angle of he sector (approx.) is :

Options

A. `65^(@)27'16''`
B. `68^(@)18'19''`
C. `56^(@)52'18''`
D. none of these

Answer

To solve the problem, we need to find the angle of a sector of a circle given that the perimeter of the sector is equal to the length of the arc of a semicircle with the same radius.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let the radius of the circle be \( r \).
- Let the angle of the sector be \( \theta \) (in degrees).

2. **Calculate the Perimeter of the Sector**:
- The perimeter \( P \) of the sector is given by the formula:
\[ P = l + 2r \]
where \( l \) is the length of the arc of the sector.

3. **Calculate the Length of the Arc of the Sector**:
- The length of the arc \( l \) can be calculated using the formula:
\[ l = \frac{\theta}{360} \times 2\pi r \]
Therefore, substituting this into the perimeter formula gives:
\[ P = \frac{\theta}{360} \times 2\pi r + 2r \]

4. **Calculate the Length of the Arc of the Semicircle**:
- The length of the arc of a semicircle with radius \( r \) is:
\[ \text{Length of semicircle arc} = \pi r \]

5. **Set the Perimeter of the Sector Equal to the Length of the Semicircle Arc**:
- According to the problem, we have:
\[ \frac{\theta}{360} \times 2\pi r + 2r = \pi r \]

6. **Simplify the Equation**:
- Rearranging gives:
\[ \frac{\theta}{360} \times 2\pi r = \pi r - 2r \]
- Factoring out \( r \) from the right side:
\[ \frac{\theta}{360} \times 2\pi r = r(\pi - 2) \]
- Dividing both sides by \( r \) (assuming \( r \neq 0 \)):
\[ \frac{\theta}{360} \times 2\pi = \pi - 2 \]

7. **Solve for \( \theta \)**:
- Rearranging gives:
\[ \theta = \frac{(\pi - 2) \times 360}{2\pi} \]
- Simplifying further:
\[ \theta = \frac{360}{2} \left(1 - \frac{2}{\pi}\right) = 180 \left(1 - \frac{2}{\pi}\right) \]

8. **Calculate the Value of \( \theta \)**:
- Using the approximate value of \( \pi \approx 3.14 \):
\[ \theta \approx 180 \left(1 - \frac{2}{3.14}\right) \approx 180 \left(1 - 0.6364\right) \approx 180 \times 0.3636 \approx 65.4545 \]
- Therefore, \( \theta \approx 65.45^\circ \).

9. **Convert to Degrees, Minutes, and Seconds**:
- The whole number part is \( 65^\circ \).
- The decimal part \( 0.4545 \times 60 \approx 27.27 \) minutes, which is approximately \( 27 \) minutes.
- The decimal part \( 0.27 \times 60 \approx 16.2 \) seconds, which is approximately \( 16 \) seconds.
- Thus, \( \theta \approx 65^\circ 27' 16'' \).

### Final Answer:
The angle of the sector is approximately \( 65^\circ 27' 16'' \).