How many sides does a regular polygon have, whose interior angle is eight times its exterior angle ?

Options

A. 16
B. 24
C. 18
D. 20

Answer

To find the number of sides of a regular polygon whose interior angle is eight times its exterior angle, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between interior and exterior angles:**
- Let the exterior angle be \( E \).
- According to the problem, the interior angle \( I \) is eight times the exterior angle:
\[ I = 8E \]

2. **Use the relationship between interior and exterior angles:**
- The sum of the interior angle and the exterior angle of a polygon is always \( 180^\circ \):
\[ I + E = 180^\circ \]
- Substituting \( I \) from the first equation:
\[ 8E + E = 180^\circ \]
- This simplifies to:
\[ 9E = 180^\circ \]
- Therefore, the exterior angle \( E \) is:
\[ E = \frac{180^\circ}{9} = 20^\circ \]

3. **Find the number of sides \( n \) of the polygon:**
- The formula for the exterior angle of a regular polygon is given by:
\[ E = \frac{360^\circ}{n} \]
- Setting the two expressions for \( E \) equal to each other:
\[ \frac{360^\circ}{n} = 20^\circ \]
- Cross-multiplying gives:
\[ 360 = 20n \]
- Dividing both sides by 20:
\[ n = \frac{360}{20} = 18 \]

4. **Conclusion:**
- The regular polygon has \( 18 \) sides.

### Final Answer:
The regular polygon has **18 sides**.