Let the number of sides of the regular polygon be \( n \).
The formula for the interior angle \( I \) of a regular polygon with \( n \) sides is given by:
\[ I = \frac{(n-2) \times 180}{n} \]
The formula for the exterior angle \( E \) of a regular polygon is:
\[ E = \frac{360}{n} \]
According to the problem, each interior angle is greater than each exterior angle by 90 degrees:
\[ I = E + 90 \]
Substituting the formulas for the interior and exterior angles into this equation gives:
\[ \frac{(n-2) \times 180}{n} = \frac{360}{n} + 90 \]
To eliminate the fractions, multiply both sides by \( n \):
\[ (n-2) \times 180 = 360 + 90n \]
Expanding the left-hand side:
\[ 180n - 360 = 360 + 90n \]
Now, let's isolate \( n \):
\[ 180n - 90n = 360 + 360 \]
This simplifies to:
\[ 90n = 720 \]
Dividing both sides by 90 gives:
\[ n = 8 \]
Thus, the polygon is an octagon (a regular polygon with 8 sides).
To verify, we can calculate the interior and exterior angles for \( n = 8 \):
- Calculate the interior angle:
\[ I = \frac{(8-2) \times 180}{8} = \frac{6 \times 180}{8} = \frac{1080}{8} = 135 \text{ degrees} \]
- Calculate the exterior angle:
\[ E = \frac{360}{8} = 45 \text{ degrees} \]
Now check the relationship:
\[ I - E = 135 - 45 = 90 \text{ degrees} \]
This confirms our solution is correct. Therefore, the number of sides in the polygon is
\[ \boxed{8} \]
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