Area of a sector of a circle is `(1)/(6)` of the circle. Find the degree measure of it minor arc.

Options

A. `90^(@)`
B. `60^(@)`
C. `45^(@)`
D. `30^(@)`

Answer

To find the degree measure of the minor arc of a sector of a circle where the area of the sector is \( \frac{1}{6} \) of the area of the entire circle, we can follow these steps:

### Step-by-step Solution:

1. **Understand the Area of a Circle and Sector**:
- The area of a circle is given by the formula:
\[ \text{Area of Circle} = \pi r^2 \]
- The area of a sector of a circle is given by the formula:
\[ \text{Area of Sector} = \frac{\pi r^2 \theta}{360} \]
where \( \theta \) is the angle in degrees.

2. **Set Up the Equation**:
- According to the problem, the area of the sector is \( \frac{1}{6} \) of the area of the circle:
\[ \frac{\pi r^2 \theta}{360} = \frac{1}{6} \cdot \pi r^2 \]

3. **Cancel Common Terms**:
- We can cancel \( \pi r^2 \) from both sides (assuming \( r \neq 0 \)):
\[ \frac{\theta}{360} = \frac{1}{6} \]

4. **Solve for \( \theta \)**:
- To find \( \theta \), multiply both sides by 360:
\[ \theta = 360 \cdot \frac{1}{6} \]
- Calculate \( \theta \):
\[ \theta = 60 \text{ degrees} \]

5. **Determine the Minor Arc**:
- The minor arc corresponds to the angle \( \theta \) of the sector, which we have found to be 60 degrees.

### Final Answer:
The degree measure of the minor arc is \( 60 \) degrees.

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