Answer:
To find the surface area of a square pyramid, you need to find the area of the base and the area of the four triangular faces, and then add them together.
First, let's find the length of one side of the base of the pyramid:
Perimeter of the square base = 4 x side length
48 inches = 4 x side length
Side length = 12 inches
Now we can find the area of the base:
Area of the base = side length squared
Area of the base = 12 inches x 12 inches
Area of the base = 144 square inches
Next, we need to find the area of each triangular face. To do this, we need to find the length of the slant height of the pyramid. We can use the Pythagorean theorem to do this:
Slant height squared = height squared + (1/2 base length) squared
Slant height squared = 8 inches squared + (6 inches) squared
Slant height squared = 64 inches squared + 36 inches squared
Slant height = square root of (64 + 36) inches
Slant height = 10 inches
Now we can find the area of each triangular face:
Area of a triangular face = (1/2 base length) x slant height
Area of a triangular face = (1/2 x 12 inches) x 10 inches
Area of a triangular face = 60 square inches
Finally, we can add the area of the base and the area of the four triangular faces together to find the total surface area of the pyramid:
Total surface area = Area of the base + (4 x Area of a triangular face)
Total surface area = 144 square inches + (4 x 60 square inches)
Total surface area = 384 square inches
Therefore, the surface area of the pyramid is 384 square inches.