The ratio between the interior angle and the exterior angle of a regular polygon is 2: 1. Find :
(i) each exterior angle of the polygon.
(ii) number of sides in the polygon.

Answer

To solve the problem, we need to find the exterior angle of the polygon and the number of sides in the polygon given that the ratio of the interior angle to the exterior angle is 2:1.

### Step-by-step Solution:

1. **Understanding the Ratio**:
We know that the ratio of the interior angle (I) to the exterior angle (E) of a polygon is given as 2:1. This can be expressed mathematically as:
\[ \frac{I}{E} = \frac{2}{1} \]

2. **Formulas for Interior and Exterior Angles**:
The formulas for the interior angle and exterior angle of a regular polygon are:
- Interior angle \( I = \frac{(n - 2) \times 180}{n} \)
- Exterior angle \( E = \frac{360}{n} \)
where \( n \) is the number of sides of the polygon.

3. **Setting Up the Equation**:
From the ratio, we can substitute the formulas into the equation:
\[ \frac{\frac{(n - 2) \times 180}{n}}{\frac{360}{n}} = \frac{2}{1} \]

4. **Simplifying the Equation**:
We can simplify the left side:
\[ \frac{(n - 2) \times 180}{360} = \frac{2}{1} \]
Now, we can simplify \( \frac{180}{360} \) to \( \frac{1}{2} \):
\[ \frac{(n - 2)}{2} = \frac{2}{1} \]

5. **Cross-Multiplying**:
Cross-multiplying gives us:
\[ n - 2 = 4 \]

6. **Solving for \( n \)**:
Adding 2 to both sides:
\[ n = 6 \]
Thus, the number of sides in the polygon is 6.

7. **Finding Each Exterior Angle**:
Now, we can find the exterior angle using the formula:
\[ E = \frac{360}{n} \]
Substituting \( n = 6 \):
\[ E = \frac{360}{6} = 60 \text{ degrees} \]

### Final Answers:
(i) Each exterior angle of the polygon is **60 degrees**.
(ii) The number of sides in the polygon is **6**.