If one of the interior angles of a regular polygon is found to be equal to `(9/8)` times of one of the interior angles of a regular hexagon, then the number of sides of the polygon is

Options

A. 4
B. 5
C. 1
D. 8

Answer

To solve the problem, we need to determine the number of sides of a regular polygon given that one of its interior angles is equal to \( \frac{9}{8} \) times one of the interior angles of a regular hexagon.

### Step-by-Step Solution:

1. **Find the interior angle of a regular hexagon:**
The formula for the interior angle \( A \) of a regular polygon with \( n \) sides is given by:
\[ A = \frac{(n-2) \times 180}{n} \]
For a hexagon, \( n = 6 \):
\[ A_{hexagon} = \frac{(6-2) \times 180}{6} = \frac{4 \times 180}{6} = \frac{720}{6} = 120 \text{ degrees} \]

2. **Calculate the interior angle of the regular polygon:**
According to the problem, the interior angle of the regular polygon \( A_{polygon} \) is:
\[ A_{polygon} = \frac{9}{8} \times A_{hexagon} = \frac{9}{8} \times 120 = \frac{1080}{8} = 135 \text{ degrees} \]

3. **Determine the exterior angle of the polygon:**
The relationship between the interior angle and the exterior angle \( E \) of a polygon is:
\[ A + E = 180 \]
Therefore, the exterior angle can be calculated as:
\[ E = 180 - A_{polygon} = 180 - 135 = 45 \text{ degrees} \]

4. **Find the number of sides of the polygon:**
The sum of the exterior angles of any polygon is always \( 360 \) degrees. The number of sides \( n \) of the polygon can be calculated using:
\[ n = \frac{360}{E} \]
Substituting the value of the exterior angle:
\[ n = \frac{360}{45} = 8 \]

Thus, the number of sides of the regular polygon is **8**.