If the area of a sector of a circle is 1/6 of area of the circle, then the angle of the sector is ________.

Answer

To solve the problem step by step, we will use the relationship between the area of a sector and the area of the whole circle.

### Step 1: Understand the formulas
The area of a circle is given by the formula:
\[ \text{Area of Circle} = \pi r^2 \]
where \( r \) is the radius of the circle.

The area of a sector of a circle is given by the formula:
\[ \text{Area of Sector} = \frac{\theta}{360} \times \text{Area of Circle} \]
where \( \theta \) is the angle of the sector in degrees.

### Step 2: Set up the equation
According to the problem, the area of the sector is \( \frac{1}{6} \) of the area of the circle. We can write this as:
\[ \text{Area of Sector} = \frac{1}{6} \times \text{Area of Circle} \]

### Step 3: Substitute the area of the circle into the equation
Substituting the area of the circle into the equation for the area of the sector, we have:
\[ \frac{\theta}{360} \times \pi r^2 = \frac{1}{6} \times \pi r^2 \]

### Step 4: Cancel out common terms
Since \( \pi r^2 \) appears on both sides of the equation, we can cancel it out (assuming \( r \neq 0 \)):
\[ \frac{\theta}{360} = \frac{1}{6} \]

### Step 5: Solve for \( \theta \)
To find \( \theta \), we can multiply both sides by 360:
\[ \theta = 360 \times \frac{1}{6} \]
Calculating the right side:
\[ \theta = 60 \]

### Conclusion
The angle of the sector is \( 60 \) degrees.

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