The net of a square-based pyramid has a perimeter of 208 cm.

Draw a possible net for this pyramid and find its corresponding total surface area.

To solve the problem, we need to understand the net of a square-based pyramid and how we can use the given perimeter to find the surface area.

### Step 1: Understand the Net of a Square-Based Pyramid
A square-based pyramid consists of:
- A square base
- Four triangular faces that connect to each side of the square base

The net of the pyramid, when unfolded, includes one square (the base) and four triangles (the sides of the pyramid). 

### Step 2: Interpret the Given Perimeter
The perimeter of the net includes the perimeter of the square base and the perimeter of the four triangles' sides. Let’s define:
- \( s \) as the side length of the square base
- \( l \) as the slant height of the triangular faces

The perimeter is made up of:
1. The four sides of the square (which is \( 4s \)),
2. The remaining two sides of each triangle (not including the base) — meaning \( 4l \) (since each triangle has two additional sides connecting to the square base).

Given that the total perimeter of the net is 208 cm:
\[
4s + 4l = 208 \quad \Rightarrow \quad s + l = 52 \quad \Rightarrow \quad l = 52 - s
\]

### Step 3: Surface Area of the Pyramid
The total surface area is the sum of the areas of the square base and the four triangular faces:
1. **Area of the square base**: \( A_{\text{base}} = s^2 \)
2. **Area of the triangular faces**: Each triangle has a base \( s \) and height \( l \), so the area of one triangle is \( \frac{1}{2} \times s \times l \). The total area for four triangles is:
   \[
   A_{\text{triangles}} = 4 \times \left( \frac{1}{2} \times s \times l \right) = 2sl
   \]

Thus, the total surface area \( A_{\text{total}} \) is:
\[
A_{\text{total}} = s^2 + 2sl
\]

### Step 4: Substitute \( l = 52 - s \) into the Surface Area Formula
Substitute \( l = 52 - s \) into the formula for surface area:
\[
A_{\text{total}} = s^2 + 2s(52 - s) = s^2 + 104s - 2s^2
\]
\[
A_{\text{total}} = -s^2 + 104s
\]

### Step 5: Find the Optimal Value of \( s \)
The value of \( s \) can be any reasonable value such that both \( s \) and \( l \) are positive. Let’s test a possible value, such as \( s = 26 \) cm (since \( s + l = 52 \)).

For \( s = 26 \):
- \( l = 52 - 26 = 26 \)
- The total surface area becomes:
  \[
  A_{\text{total}} = -26^2 + 104 \times 26 = -676 + 2704 = 2028 \, \text{cm}^2
  \]

So, the total surface area is 2028 cm².

I can now create a net for you! Would you like to see a possible net drawing for the pyramid? Also, here are some follow-up questions to explore:

### Follow-up Questions:
1. How would the surface area change if the slant height was increased by 10 cm?
2. Can you calculate the volume of the square-based pyramid if its height is given?
3. What happens to the total surface area if the side length of the base is doubled?
4. How do you calculate the lateral surface area of a pyramid?
5. How would the surface area change if the perimeter of the net was 300 cm?

**Tip**: When solving for surface area, always break down complex shapes into simple components like squares and triangles.

The net of a square-based pyramid has a perimeter of 208 cm. Draw a possible net for this pyramid and find its corresponding total surface area.

Learn how to calculate the total surface area of a square-based pyramid with a given net perimeter of 208 cm. This problem explores geometric concepts such as perimeter, surface area, and includes step-by-step calculations for the square base and triangular faces. Suitable for students in Grades 8-10.

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