If the perimeter of a certain sector of a circle is equal to the length of the arc of the semicircle having the same radius , find the angle of the sector in degrees .

Answer

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

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a sector of a circle and a semicircle with the same radius. We need to find the angle of the sector (let's denote it as \( \theta \)) in degrees, given that the perimeter of the sector is equal to the length of the arc of the semicircle.

2. **Formulas Involved**:
- The length of the arc of a semicircle with radius \( r \) is given by:
\[ \text{Length of arc of semicircle} = \frac{1}{2} \times 2\pi r = \pi r \]
- The perimeter of the sector is given by the formula:
\[ \text{Perimeter of sector} = \text{Length of arc} + 2r \]
- The length of the arc of the sector can be expressed as:
\[ \text{Length of arc of sector} = \frac{\theta}{360} \times 2\pi r = \frac{\theta \pi r}{180} \]

3. **Set Up the Equation**:
According to the problem, the perimeter of the sector is equal to the length of the arc of the semicircle:
\[ \frac{\theta \pi r}{180} + 2r = \pi r \]

4. **Simplify the Equation**:
We can cancel \( r \) from both sides (assuming \( r \neq 0 \)):
\[ \frac{\theta \pi}{180} + 2 = \pi \]
Rearranging gives:
\[ \frac{\theta \pi}{180} = \pi - 2 \]

5. **Solve for \( \theta \)**:
Multiply both sides by \( 180/\pi \):
\[ \theta = (180/\pi)(\pi - 2) \]
Simplifying this gives:
\[ \theta = 180 - \frac{360}{\pi} \]

6. **Calculate the Value of \( \theta \)**:
Using \( \pi \approx 3.14 \):
\[ \theta \approx 180 - \frac{360}{3.14} \approx 180 - 114.59 \approx 65.41 \text{ degrees} \]

### Final Answer:
The angle of the sector \( \theta \) is approximately \( 65.41 \) degrees.