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The problem involves calculating the surface area of a square-based pyramid where all the triangular faces are identical isosceles triangles. The dimensions given are:

- The base of each triangular face is \( 9.2 \, \text{cm} \).
- The slant height (height of each triangular face) is \( 14 \, \text{cm} \).

### Steps to calculate the surface area:

1. **Calculate the area of the square base**:
   - The base of the pyramid is a square with side lengths equal to \( 9.2 \, \text{cm} \).
   - The area \( A_{\text{base}} \) of the square is:
   \[
   A_{\text{base}} = 9.2^2 = 84.64 \, \text{cm}^2
   \]

2. **Calculate the area of one triangular face**:
   - The area \( A_{\text{triangle}} \) of each triangular face is given by:
   \[
   A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
   \]
   - Here, the base of the triangle is \( 9.2 \, \text{cm} \) and the height (slant height) is \( 14 \, \text{cm} \):
   \[
   A_{\text{triangle}} = \frac{1}{2} \times 9.2 \times 14 = 64.4 \, \text{cm}^2
   \]

3. **Calculate the total area of the four triangular faces**:
   - Since the pyramid has four identical triangular faces, the total area for these faces is:
   \[
   A_{\text{triangular faces}} = 4 \times 64.4 = 257.6 \, \text{cm}^2
   \]

4. **Calculate the total surface area of the pyramid**:
   - The total surface area \( A_{\text{total}} \) is the sum of the area of the square base and the area of the four triangular faces:
   \[
   A_{\text{total}} = A_{\text{base}} + A_{\text{triangular faces}} = 84.64 + 257.6 = 342.24 \, \text{cm}^2
   \]

5. **Round the answer**:
   - Rounding \( 342.24 \, \text{cm}^2 \) to the nearest whole number, the surface area of the pyramid is:
   \[
   \boxed{342 \, \text{cm}^2}
   \]

Would you like more details on any of these steps or have any further questions? 

### Related Questions:
1. How do you calculate the volume of a square-based pyramid?
2. What is the formula for the surface area of a pyramid with different triangular faces?
3. How does the slant height differ from the perpendicular height of a pyramid?
4. What is the importance of rounding in practical applications of surface area calculations?
5. How would the surface area change if the base of the pyramid was a rectangle instead of a square?

### Tip:
Always double-check units and rounding, especially when dealing with real-world applications or exams!

The triangular faces of the square-based pyramid below are all identical isosceles triangles. Work out the surface area of this pyramid. Round your answer to the nearest cm².

Learn how to calculate the surface area of a square-based pyramid with identical isosceles triangular faces. Using a step-by-step process, this problem covers key geometry concepts and applies formulas for the area of squares and triangles. Suitable for students in Grades 8-10.

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