The exterior angle of a regular polygon is one-third of its interior angle. Find the number of sides in the polygon.

Answer

To solve the problem, we need to find the number of sides in a regular polygon where the exterior angle is one-third of its interior angle. We can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The formula for the interior angle (I) of a regular polygon with n sides is:
\[ I = \frac{(n - 2) \times 180}{n} \]
- The formula for the exterior angle (E) of a regular polygon with n sides is:
\[ E = \frac{360}{n} \]

2. **Set Up the Equation**:
- According to the problem, the exterior angle is one-third of the interior angle:
\[ E = \frac{1}{3} I \]

3. **Substitute the Formulas into the Equation**:
- Replace E and I with their respective formulas:
\[ \frac{360}{n} = \frac{1}{3} \left( \frac{(n - 2) \times 180}{n} \right) \]

4. **Simplify the Equation**:
- Multiply both sides by n to eliminate the denominator:
\[ 360 = \frac{1}{3} (n - 2) \times 180 \]
- Multiply both sides by 3 to eliminate the fraction:
\[ 1080 = (n - 2) \times 180 \]

5. **Divide Both Sides by 180**:
- Simplifying gives:
\[ \frac{1080}{180} = n - 2 \]
- This simplifies to:
\[ 6 = n - 2 \]

6. **Solve for n**:
- Add 2 to both sides:
\[ n = 6 + 2 \]
- Therefore:
\[ n = 8 \]

7. **Conclusion**:
- The number of sides in the polygon is 8.