The ratio between the exterior angle and the interior angle of a regular polygon is `1:3`. Find the number of the sides of the polygon.

Options

A. `12`
B. `6`
C. `8`
D. `10`

Explanation

To solve the problem of finding the number of sides of a regular polygon given the ratio of its exterior angle to its interior angle is 1:3, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Angles**:
Let the exterior angle be \( x \). According to the problem, the interior angle will then be \( 3x \) since the ratio of the exterior angle to the interior angle is 1:3.

2. **Use the Angle Relationship**:
The sum of the exterior angle and the interior angle of a polygon is always \( 180^\circ \). Therefore, we can write the equation:
\[ x + 3x = 180^\circ \]

3. **Combine Like Terms**:
Simplifying the equation gives:
\[ 4x = 180^\circ \]

4. **Solve for \( x \)**:
Divide both sides by 4 to find \( x \):
\[ x = \frac{180^\circ}{4} = 45^\circ \]

5. **Find the Number of Sides**:
The exterior angle of a regular polygon can also be calculated using the formula:
\[ \text{Exterior Angle} = \frac{360^\circ}{n} \]
where \( n \) is the number of sides. Setting this equal to \( x \):
\[ \frac{360^\circ}{n} = 45^\circ \]

6. **Solve for \( n \)**:
Rearranging gives:
\[ n = \frac{360^\circ}{45^\circ} \]
Calculating this gives:
\[ n = 8 \]

7. **Conclusion**:
Therefore, the number of sides of the polygon is \( 8 \).

### Final Answer:
The polygon has **8 sides**.