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To find the total surface area of the pyramid, we need to calculate both the area of its square base and the area of its four triangular sides (lateral faces).

### Given Data
1. **Base side length** = 10 cm
2. **Slant height** (height of each triangular side) = 11.2 cm

### Steps to Calculate the Surface Area

1. **Area of the Base (Square)**
   \[
   \text{Area of base} = \text{side}^2 = 10^2 = 100 \text{ cm}^2
   \]

2. **Area of One Triangular Side**
   Each triangular side has a base of 10 cm (the side of the square base) and a slant height of 11.2 cm.
   \[
   \text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 10 \times 11.2 = 56 \text{ cm}^2
   \]

3. **Total Area of Four Triangular Sides**
   \[
   \text{Total area of triangles} = 4 \times 56 = 224 \text{ cm}^2
   \]

4. **Total Surface Area of the Pyramid**
   \[
   \text{Total surface area} = \text{Area of base} + \text{Total area of triangles} = 100 + 224 = 324 \text{ cm}^2
   \]

### Final Answer
The total surface area of the pyramid is **324 cm²** (rounded to the nearest square centimeter).

Would you like further details or have any questions?

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Here are some related questions that might help deepen your understanding:

1. How would the total surface area change if the slant height were increased?
2. What if the base were a different polygon, such as a triangle or hexagon?
3. How can we calculate the surface area if we only know the height of the pyramid, not the slant height?
4. How does the slant height relate to the actual height of the pyramid?
5. What would be the volume of this pyramid?

**Tip:** For pyramids, the slant height is key for surface area calculations since it directly determines the area of the triangular faces.

Find the total surface area of the pyramid shown. The base is a square. Round to the nearest square cm.

Learn how to find the total surface area of a pyramid with a square base and slant height. This problem provides step-by-step calculations using geometric formulas, suitable for students in Grades 7-9.

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