The sum of the interior angles of a polygon is four times the sum of its exterior angles. Find the number of sides in the polygon.

Answer

To solve the problem, we need to find the number of sides in a polygon where the sum of the interior angles is four times the sum of its exterior angles.

### Step-by-Step Solution:

1. **Understand the formulas for angles in a polygon:**
- The sum of the interior angles of a polygon with \( n \) sides is given by the formula:
\[ \text{Sum of interior angles} = (n - 2) \times 180 \]
- The sum of the exterior angles of any polygon is always:
\[ \text{Sum of exterior angles} = 360 \text{ degrees} \]

2. **Set up the equation based on the problem statement:**
- According to the problem, the sum of the interior angles is four times the sum of the exterior angles:
\[ (n - 2) \times 180 = 4 \times 360 \]

3. **Calculate the right-hand side:**
- Calculate \( 4 \times 360 \):
\[ 4 \times 360 = 1440 \]
- Now, we can rewrite the equation:
\[ (n - 2) \times 180 = 1440 \]

4. **Solve for \( n \):**
- Divide both sides by 180 to isolate \( n - 2 \):
\[ n - 2 = \frac{1440}{180} \]
- Simplifying the right side:
\[ n - 2 = 8 \]
- Now, add 2 to both sides:
\[ n = 8 + 2 = 10 \]

5. **Conclusion:**
- The number of sides in the polygon is \( n = 10 \).