Polygon Angle Difference Calculation

To find the number of sides ($n$) of a polygon, we use the formulas for its interior and exterior angles.

• Interior Angle ($I$): $I = \frac{(n-2) \times 180^\circ}{n}$
• Exterior Angle ($E$): $E = \frac{360^\circ}{n}$

We are given that the difference between the interior and exterior angles is $36^\circ$.

Given: $I - E = 36^\circ$

Step-by-Step Calculation

1. Substitute the formulas for $I$ and $E$ into the given equation:

   $ \frac{(n-2) \times 180^\circ}{n} - \frac{360^\circ}{n} = 36^\circ $

2. Simplify the left side of the equation. Since the denominators are the same, combine the numerators:

   $ \frac{(n-2) \times 180^\circ - 360^\circ}{n} = 36^\circ $

3. Expand the term in the numerator:

   $ \frac{180^\circ n - 360^\circ - 360^\circ}{n} = 36^\circ $

4. Combine the constant terms in the numerator:

   $ \frac{180^\circ n - 720^\circ}{n} = 36^\circ $

5. Multiply both sides by $n$ to eliminate the denominator:

   $ 180^\circ n - 720^\circ = 36^\circ n $

6. Rearrange the equation to group terms with $n$:

   $ 180^\circ n - 36^\circ n = 720^\circ $

7. Combine the terms with $n$:

   $ 144^\circ n = 720^\circ $

8. Solve for $n$ by dividing both sides by $144^\circ$:

   $ n = \frac{720^\circ}{144^\circ} $

   $ n = 5 $

Therefore, the polygon has 5 sides.