If the measure of the interior angle of a regular polygon is `108^@`, then how many sides does it have?

Options

A. 6
B. 8
C. 9
D. 5

Answer

To find the number of sides of a regular polygon given that the measure of its interior angle is \(108^\circ\), we can follow these steps:

### Step 1: Use the formula for the interior angle of a regular polygon.
The formula for the measure of an interior angle of a regular polygon with \(n\) sides is given by:
\[ \text{Interior Angle} = \frac{(n-2) \times 180}{n} \]
Given that the interior angle is \(108^\circ\), we can set up the equation:
\[ \frac{(n-2) \times 180}{n} = 108 \]

### Step 2: Multiply both sides by \(n\) to eliminate the fraction.
\[ (n-2) \times 180 = 108n \]

### Step 3: Distribute \(180\) on the left side.
\[ 180n - 360 = 108n \]

### Step 4: Rearrange the equation to isolate terms involving \(n\).
Subtract \(108n\) from both sides:
\[ 180n - 108n - 360 = 0 \]
This simplifies to:
\[ 72n - 360 = 0 \]

### Step 5: Solve for \(n\).
Add \(360\) to both sides:
\[ 72n = 360 \]
Now, divide both sides by \(72\):
\[ n = \frac{360}{72} = 5 \]

### Conclusion:
The regular polygon has \(5\) sides.