To find the surface area of a square pyramid, we need to calculate both the area of the square base and the area of the four triangular faces. Here's how to break it down:

1. Area of the Square Base:

The area $A_{\text{base}}$ of the square base is given by: $A_{\text{base}} = \text{side length}^2$ The side length is 5 ft, so: $A_{\text{base}} = 5^2 = 25 \, \text{ft}^2$

2. Area of the Triangular Faces:

Each of the four triangular faces has a base of 5 ft (the side length of the square base) and a slant height of 6 ft. The area $A_{\text{triangle}}$ of one triangular face is: $A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{slant height}$ $A_{\text{triangle}} = \frac{1}{2} \times 5 \times 6 = 15 \, \text{ft}^2$

Since there are 4 triangular faces, the total area of the triangular faces is: $A_{\text{triangles total}} = 4 \times 15 = 60 \, \text{ft}^2$

3. Total Surface Area:

Now, we add the area of the square base and the total area of the triangular faces: $A_{\text{total}} = A_{\text{base}} + A_{\text{triangles total}} = 25 + 60 = 85 \, \text{ft}^2$

Final Answer:

The surface area of the square pyramid is 85 ft².

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Would you like further details or explanations on any step? Here are some additional questions:

1. How would the surface area change if the side length was 6 ft instead of 5 ft?
2. How would you calculate the volume of the square pyramid?
3. If the slant height was different, how would you adjust the calculation for the surface area?
4. Can you use the same formula for other pyramids with triangular bases or different shapes?
5. How can you derive the slant height from other known measurements?

Tip: Always ensure that the slant height is perpendicular to the base when calculating the area of the triangular faces!