Find the number of sides of a regular polygon whose each exterior angle has a measure of `45^@`.

Answer

To find the number of sides of a regular polygon whose each exterior angle measures \(45^\circ\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between exterior angles and number of sides**:
The measure of each exterior angle of a regular polygon can be calculated using the formula:
\[ \text{Exterior Angle} = \frac{360^\circ}{n} \]
where \(n\) is the number of sides of the polygon.

2. **Set up the equation**:
Since we know the measure of the exterior angle is \(45^\circ\), we can substitute this value into the formula:
\[ 45^\circ = \frac{360^\circ}{n} \]

3. **Cross-multiply to solve for \(n\)**:
To eliminate the fraction, we can cross-multiply:
\[ 45^\circ \cdot n = 360^\circ \]

4. **Isolate \(n\)**:
Now, divide both sides of the equation by \(45^\circ\) to solve for \(n\):
\[ n = \frac{360^\circ}{45^\circ} \]

5. **Simplify the fraction**:
We can simplify the right side:
\[ n = \frac{360}{45} \]
To simplify, we can divide both the numerator and the denominator by \(5\):
\[ n = \frac{72}{9} = 8 \]

6. **Conclusion**:
Therefore, the number of sides \(n\) of the regular polygon is \(8\).

### Final Answer:
The number of sides of the regular polygon is \(8\).

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