The sum of the interior angles of a polygon is five 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 \( n \) in a polygon given that the sum of its interior angles is five times the sum of its exterior angles.

### Step-by-Step Solution:

1. **Understand the Formulas**:
- 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:
\[ \text{Sum of exterior angles} = 360 \]

2. **Set Up the Equation**:
- According to the problem, the sum of the interior angles is five times the sum of the exterior angles. Therefore, we can write the equation:
\[ (n - 2) \times 180 = 5 \times 360 \]

3. **Simplify the Right Side**:
- Calculate \( 5 \times 360 \):
\[ 5 \times 360 = 1800 \]
- Now, we can rewrite our equation:
\[ (n - 2) \times 180 = 1800 \]

4. **Divide Both Sides by 180**:
- To isolate \( n - 2 \), divide both sides of the equation by 180:
\[ n - 2 = \frac{1800}{180} \]
- Simplifying gives:
\[ n - 2 = 10 \]

5. **Solve for \( n \)**:
- Now, add 2 to both sides to find \( n \):
\[ n = 10 + 2 = 12 \]

6. **Conclusion**:
- Therefore, the number of sides in the polygon is:
\[ \boxed{12} \]