The interior angles of a polygon are in arithmetic progression. The smallest angle is `52^(@)` and the common difference is `8^(@)` . Find the number of sides of the polygon.

Answer

To solve the problem, we need to find the number of sides of a polygon whose interior angles are in arithmetic progression. The smallest angle is \(52^\circ\) and the common difference is \(8^\circ\).

### Step-by-Step Solution:

1. **Identify the first term and common difference**:
- Let the smallest angle (first term) be \(A = 52^\circ\).
- Let the common difference be \(d = 8^\circ\).

2. **Express the interior angles**:
- The interior angles of the polygon can be expressed as:
\[ A, A + d, A + 2d, \ldots, A + (n-1)d \]
- This means the angles are:
\[ 52^\circ, 52^\circ + 8^\circ, 52^\circ + 2 \cdot 8^\circ, \ldots, 52^\circ + (n-1) \cdot 8^\circ \]

3. **Sum of the interior angles**:
- The sum of the interior angles of a polygon with \(n\) sides is given by:
\[ S = (n-2) \times 180^\circ \]

4. **Sum of the angles in arithmetic progression**:
- The sum of the angles can also be calculated using the formula for the sum of an arithmetic series:
\[ S = \frac{n}{2} \times (2A + (n-1)d) \]
- Substituting \(A = 52^\circ\) and \(d = 8^\circ\):
\[ S = \frac{n}{2} \times (2 \cdot 52 + (n-1) \cdot 8) \]

5. **Set the two expressions for the sum equal**:
- Equating the two expressions for \(S\):
\[ \frac{n}{2} \times (2 \cdot 52 + (n-1) \cdot 8) = (n-2) \times 180 \]

6. **Simplify the equation**:
- This simplifies to:
\[ \frac{n}{2} \times (104 + 8n - 8) = (n-2) \times 180 \]
- Which further simplifies to:
\[ \frac{n}{2} \times (8n + 96) = (n-2) \times 180 \]
- Multiplying both sides by 2 to eliminate the fraction:
\[ n(8n + 96) = 2(n-2) \times 180 \]
- Expanding both sides:
\[ 8n^2 + 96n = 360n - 720 \]

7. **Rearranging the equation**:
- Rearranging gives:
\[ 8n^2 + 96n - 360n + 720 = 0 \]
- Which simplifies to:
\[ 8n^2 - 264n + 720 = 0 \]
- Dividing the entire equation by 8:
\[ n^2 - 33n + 90 = 0 \]

8. **Factoring the quadratic equation**:
- The quadratic can be factored as:
\[ (n - 30)(n - 3) = 0 \]
- Thus, the solutions for \(n\) are:
\[ n = 30 \quad \text{or} \quad n = 3 \]

9. **Determine the valid solution**:
- We need to check which value of \(n\) is valid. For \(n = 30\):
- The largest angle would be \(52 + 29 \cdot 8 = 284^\circ\), which is not possible for a polygon.
- Therefore, the only valid solution is:
\[ n = 3 \]

### Conclusion:
The number of sides of the polygon is **3**.