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 out how many sides a regular polygon has if each of its interior angles is \(165^\circ\), we can follow these steps:

### Step 1: Understand the relationship between the interior angle and the number of sides
The formula for the sum of the interior angles of a polygon with \(n\) sides is given by:
\[ \text{Sum of interior angles} = (n - 2) \times 180 \]
Since all interior angles are equal in a regular polygon, each interior angle can be calculated as:
\[ \text{Each interior angle} = \frac{\text{Sum of interior angles}}{n} = \frac{(n - 2) \times 180}{n} \]

### Step 2: Set up the equation
Given that each interior angle is \(165^\circ\), we can set up the equation:
\[ 165 = \frac{(n - 2) \times 180}{n} \]

### Step 3: Multiply both sides by \(n\) to eliminate the fraction
\[ 165n = (n - 2) \times 180 \]

### Step 4: Expand the right side
\[ 165n = 180n - 360 \]

### Step 5: Rearrange the equation to isolate \(n\)
Now, we can move all terms involving \(n\) to one side:
\[ 165n - 180n = -360 \]
\[ -15n = -360 \]

### Step 6: Solve for \(n\)
Dividing both sides by \(-15\):
\[ n = \frac{360}{15} \]
\[ n = 24 \]

### Conclusion
Thus, the regular polygon has **24 sides**.

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