The ratio between the exterior angle and the interior angle of a regular polygon is 1 : 4. Find the number of sides in the polygon.

Answer

To find the number of sides in a regular polygon where the ratio of the exterior angle to the interior angle is 1:4, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between exterior and interior angles**:
- The exterior angle (E) and the interior angle (I) of a polygon are related by the equation:
\[ E + I = 180^\circ \]
- Given the ratio \( \frac{E}{I} = \frac{1}{4} \), we can express this as:
\[ E = \frac{1}{4} I \]

2. **Express interior angle in terms of exterior angle**:
- From the equation \( E + I = 180^\circ \), substitute \( E \):
\[ \frac{1}{4} I + I = 180^\circ \]
- Combine the terms:
\[ \frac{5}{4} I = 180^\circ \]

3. **Solve for the interior angle (I)**:
- Multiply both sides by \( \frac{4}{5} \):
\[ I = 180^\circ \times \frac{4}{5} = 144^\circ \]

4. **Use the formula for the interior angle of a regular polygon**:
- The formula for the interior angle of a regular polygon with \( n \) sides is:
\[ I = \frac{(n-2) \times 180^\circ}{n} \]
- Set this equal to the interior angle we found:
\[ \frac{(n-2) \times 180^\circ}{n} = 144^\circ \]

5. **Clear the fraction by multiplying both sides by \( n \)**:
- This gives:
\[ (n-2) \times 180^\circ = 144^\circ n \]

6. **Distribute and rearrange the equation**:
- Expanding the left side:
\[ 180n - 360 = 144n \]
- Rearranging gives:
\[ 180n - 144n = 360 \]
\[ 36n = 360 \]

7. **Solve for \( n \)**:
- Divide both sides by 36:
\[ n = \frac{360}{36} = 10 \]

### Conclusion:
The number of sides in the polygon is \( n = 10 \).

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