The sum of all the interior angles of a regular polygon is four times the sum of its exterior angles. The polygon is :

Options

A. hexagon
B. triangle
C. decagon
D. nonagon

Answer

To solve the problem, we need to find out which regular polygon has the property that the sum of its interior angles is four 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:
\[ \text{Sum of Interior Angles} = (n - 2) \times 180^\circ \]
- 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 four times the sum of the exterior angles. Therefore, we can write the equation:
\[ (n - 2) \times 180 = 4 \times 360 \]

3. **Calculate the Right Side**:
First, calculate \( 4 \times 360 \):
\[ 4 \times 360 = 1440 \]
So, the equation becomes:
\[ (n - 2) \times 180 = 1440 \]

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

5. **Find \( n \)**:
Now, add 2 to both sides to find \( n \):
\[ n = 8 + 2 = 10 \]

6. **Identify the Polygon**:
Since \( n = 10 \), this means the polygon has 10 sides. A polygon with 10 sides is called a decagon.

### Conclusion:
The polygon is a **decagon**.