The interior angle of a regular polygon exceeds its exterior angle by `108^(@)`. How many sides does the polygon have ?

Options

A. 16
B. 14
C. 12
D. 10

Answer

To solve the problem, we need to find the number of sides of a regular polygon given that its interior angle exceeds its exterior angle by \(108^\circ\).

### Step-by-Step Solution:

1. **Define the Angles**:
Let the interior angle be \(x\) and the exterior angle be \(y\). According to the problem, we have:
\[ x - y = 108^\circ \]

2. **Relationship Between Interior and Exterior Angles**:
We know that the interior angle and exterior angle of a polygon are related by the equation:
\[ x + y = 180^\circ \]

3. **Set Up the Equations**:
Now we have two equations:
1. \(x - y = 108^\circ\) (Equation 1)
2. \(x + y = 180^\circ\) (Equation 2)

4. **Add the Two Equations**:
Adding Equation 1 and Equation 2:
\[ (x - y) + (x + y) = 108^\circ + 180^\circ \]
This simplifies to:
\[ 2x = 288^\circ \]

5. **Solve for \(x\)**:
Dividing both sides by 2 gives:
\[ x = 144^\circ \]

6. **Substitute to Find \(y\)**:
Now, substitute \(x\) back into Equation 2 to find \(y\):
\[ 144^\circ + y = 180^\circ \]
Solving for \(y\):
\[ y = 180^\circ - 144^\circ = 36^\circ \]

7. **Find the Number of Sides**:
The exterior angle \(y\) is related to the number of sides \(n\) of the polygon by the formula:
\[ y = \frac{360^\circ}{n} \]
Substituting \(y = 36^\circ\):
\[ 36^\circ = \frac{360^\circ}{n} \]

8. **Solve for \(n\)**:
Rearranging gives:
\[ n = \frac{360^\circ}{36^\circ} = 10 \]

### Conclusion:
The regular polygon has **10 sides**.

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