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 \( n = 8 \):
1. Calculate the interior angle:
\[
I = \frac{(8-2) \times 180}{8} = \frac{6 \times 180}{8} = \frac{1080}{8} = 135 \text{ degrees}
\]
2. 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
\[
\boxed{8}
\]