The ratio between an exterior angle and an interior angle of a regular polygon is 2 : 3. Find the number of sides in the polygon.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the relationship between exterior and interior angles
Given that the ratio between an exterior angle and an interior angle of a regular polygon is 2:3, we can denote the exterior angle as \( E \) and the interior angle as \( I \). Thus, we can express this relationship as:
\[ \frac{E}{I} = \frac{2}{3} \]

### Step 2: Express the angles in terms of \( n \)
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:
\[ I = \frac{180(n - 2)}{n} \]

### Step 3: Set up the equation using the ratio
From the ratio \( \frac{E}{I} = \frac{2}{3} \), we can substitute the expressions for \( E \) and \( I \):
\[ \frac{\frac{360}{n}}{\frac{180(n - 2)}{n}} = \frac{2}{3} \]

### Step 4: Simplify the equation
This simplifies to:
\[ \frac{360}{180(n - 2)} = \frac{2}{3} \]
\[ \frac{2}{n - 2} = \frac{2}{3} \]

### Step 5: Cross-multiply to solve for \( n \)
Cross-multiplying gives:
\[ 2 \cdot 3 = 2(n - 2) \]
\[ 6 = 2n - 4 \]

### Step 6: Rearrange the equation
Rearranging the equation to isolate \( n \):
\[ 2n = 6 + 4 \]
\[ 2n = 10 \]
\[ n = 5 \]

### Conclusion
The number of sides in the polygon is \( n = 5 \).

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