The sum of the interior angles of a regular polygon is equal to six times the sum of exterior angles. Find the number of sides of the polygon.

Answer

To solve the problem, we need to find the number of sides \( n \) of a regular polygon where the sum of the interior angles is equal to six times the sum of the exterior angles.

### Step-by-step Solution:

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

2. **Set up the equation**:
- According to the problem, the sum of the interior angles is equal to six times the sum of the exterior angles. Therefore, we can set up the equation:
\[ (n - 2) \times 180^\circ = 6 \times 360^\circ \]

3. **Calculate the right side**:
- Calculate \( 6 \times 360^\circ \):
\[ 6 \times 360^\circ = 2160^\circ \]

4. **Substitute and simplify**:
- Now substitute this value back into the equation:
\[ (n - 2) \times 180^\circ = 2160^\circ \]
- To isolate \( n - 2 \), divide both sides by \( 180^\circ \):
\[ n - 2 = \frac{2160^\circ}{180^\circ} \]

5. **Perform the division**:
- Calculate \( \frac{2160}{180} \):
\[ n - 2 = 12 \]

6. **Solve for \( n \)**:
- Now, add 2 to both sides to find \( n \):
\[ n = 12 + 2 = 14 \]

### Final Answer:
The number of sides of the polygon is \( n = 14 \).

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