If the measure of the Interior angle of a regular polygon is `90^@` greater than the measure of its exterior angle, then how many sides does it have?

Options

A. 8
B. 9
C. 10
D. 12

Answer

To solve the problem, we need to find the number of sides of a regular polygon given that the measure of its interior angle is 90 degrees greater than the measure of its exterior angle.

### Step-by-step Solution:

1. **Understand the relationship between interior and exterior angles:**
The interior angle (I) and exterior angle (E) of a regular polygon are related by the equation:
\[ I = E + 90^\circ \]

2. **Use the formulas for interior and exterior angles:**
The formula for the exterior angle of a regular polygon with \( n \) sides is:
\[ E = \frac{360^\circ}{n} \]
The formula for the interior angle is:
\[ I = \frac{180(n-2)}{n} \]

3. **Set up the equation:**
Substitute the expression for \( E \) into the relationship:
\[ \frac{180(n-2)}{n} = \frac{360}{n} + 90 \]

4. **Clear the fractions by multiplying through by \( n \):**
\[ 180(n-2) = 360 + 90n \]

5. **Distribute and simplify:**
\[ 180n - 360 = 360 + 90n \]

6. **Rearrange the equation:**
Move all terms involving \( n \) to one side and constant terms to the other:
\[ 180n - 90n = 360 + 360 \]
\[ 90n = 720 \]

7. **Solve for \( n \):**
\[ n = \frac{720}{90} = 8 \]

### Conclusion:
The regular polygon has **8 sides**.