How many sides does a regular polygon have whose interior and exterior angles are in the ratio 2 : 1?

Options

A. 3
B. 5
C. 6
D. 12

Answer

To determine how many sides a regular polygon has when the ratio of its interior and exterior angles is 2:1, we can follow these steps:

### Step 1: Define the interior and exterior angles
For a regular polygon with \( n \) sides:
- The exterior angle \( E \) is given by:
\[ E = \frac{360}{n} \]
- The interior angle \( I \) is given by:
\[ I = \frac{(n - 2) \times 180}{n} \]

### Step 2: Set up the ratio of interior to exterior angles
According to the problem, the ratio of the interior angle to the exterior angle is 2:1. Therefore, we can write:
\[ \frac{I}{E} = \frac{2}{1} \]

### Step 3: Substitute the expressions for \( I \) and \( E \)
Substituting the expressions for \( I \) and \( E \) into the ratio gives:
\[ \frac{\frac{(n - 2) \times 180}{n}}{\frac{360}{n}} = \frac{2}{1} \]

### Step 4: Simplify the equation
This simplifies to:
\[ \frac{(n - 2) \times 180}{360} = 2 \]
Cancelling \( n \) from the numerator and denominator:
\[ \frac{(n - 2) \times 180}{360} = 2 \implies \frac{(n - 2)}{2} = 2 \]

### Step 5: Solve for \( n \)
Multiplying both sides by 2 gives:
\[ n - 2 = 4 \]
Adding 2 to both sides results in:
\[ n = 6 \]

### Conclusion
Thus, the regular polygon has **6 sides**.