Question

I’m a regular polygon each interior angle is greater than each exterior angle by 90 degrees. Calculate the number of sides in the polygon.

Answer 1

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

\[
\boxed{8}
\]