The perimeter of a certain sector of a circle is equal to half that of the circle of which it is a sector. The circular measure of one angle of the sector is

Options

A. `(pi-2)` radian
B. `(pi+2)` radian
C. `pi` radian
D. `(pi-3)` radian

Answer

To solve the problem, we need to find the circular measure of the angle of a sector of a circle given that the perimeter of the sector is equal to half the perimeter of the entire circle.

### Step-by-Step Solution:

1. **Understand the Perimeter of the Circle**:
The perimeter (circumference) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

2. **Perimeter of the Sector**:
The perimeter of a sector of a circle includes the lengths of the two radii and the arc length. If \( \theta \) is the angle of the sector in radians, the arc length \( L \) can be calculated as:
\[ L = r\theta \]
Therefore, the perimeter \( P \) of the sector is:
\[ P = r\theta + 2r = r(\theta + 2) \]

3. **Setting Up the Equation**:
According to the problem, the perimeter of the sector is equal to half the perimeter of the circle:
\[ r(\theta + 2) = \frac{1}{2} \times 2\pi r \]
Simplifying this gives:
\[ r(\theta + 2) = \pi r \]

4. **Dividing by \( r \)**:
Assuming \( r \neq 0 \), we can divide both sides by \( r \):
\[ \theta + 2 = \pi \]

5. **Solving for \( \theta \)**:
Rearranging the equation gives:
\[ \theta = \pi - 2 \]

6. **Converting to Degrees**:
To convert \( \theta \) from radians to degrees, we use the conversion factor \( \frac{180}{\pi} \):
\[ \theta = (\pi - 2) \times \frac{180}{\pi} \]
Simplifying this gives:
\[ \theta = 180 - \frac{360}{\pi} \]

### Final Answer:
The circular measure of the angle of the sector is:
\[ \theta = 180 - \frac{360}{\pi} \text{ degrees} \]