How many sides does a regular polygon have if each of its interior angles is `165^@` ?

Options

A. `21`
B. `22`
C. `23`
D. `24`

Explanation

To find the number of sides \( n \) of a regular polygon given that each of its interior angles is \( 165^\circ \), we can use the formula for the interior angle of a regular polygon:

\[ \text{Interior Angle} = \frac{(n - 2) \times 180}{n} \]

where \( n \) is the number of sides.

### Step-by-Step Solution:

1. **Set up the equation**: Since we know that each interior angle is \( 165^\circ \), we can set up the equation:
\[ \frac{(n - 2) \times 180}{n} = 165 \]

**Hint**: Start with the formula for the interior angle of a polygon.

2. **Cross-multiply to eliminate the fraction**: Multiply both sides by \( n \) to eliminate the denominator:
\[ (n - 2) \times 180 = 165n \]

**Hint**: To simplify the equation, eliminate the fraction by multiplying both sides by \( n \).

3. **Distribute the \( 180 \)**: Expand the left side:
\[ 180n - 360 = 165n \]

**Hint**: Remember to distribute correctly when multiplying.

4. **Rearrange the equation**: Move all terms involving \( n \) to one side:
\[ 180n - 165n = 360 \]
This simplifies to:
\[ 15n = 360 \]

**Hint**: Combine like terms to isolate \( n \).

5. **Solve for \( n \)**: Divide both sides by \( 15 \):
\[ n = \frac{360}{15} \]

**Hint**: To find \( n \), perform the division.

6. **Calculate the value**: Simplifying \( \frac{360}{15} \):
\[ n = 24 \]

**Hint**: Perform the division step carefully.

### Final Answer:
The regular polygon has **24 sides**.