What is the surface area of a square pyramid where the base edge is 8 ft. and the slant height is 16 ft.?
To find the surface area of a square pyramid, we need to find the area of the base and the area of the four triangular faces.
The area of a square base can be found by multiplying the length of one side by itself:
Area of the base = side^2 = 8 ft. * 8 ft. = 64 ft^2.
The area of a triangle can be found by multiplying the base by the height and dividing by 2. In this case, the base of the triangle is the same as the side of the square base. The height of the triangle can be found using the Pythagorean theorem: height^2 = slant height^2 - base^2. Plugging in the given values, we get:
height^2 = 16 ft. * 16 ft. - 8 ft. * 8 ft. = 256 ft^2 - 64 ft^2 = 192 ft^2.
Taking the square root of both sides, we find the height to be square root of 192 ft^2 = 13.86 ft.
Now we can find the area of one triangular face:
Area of one triangular face = (1/2) * base * height = (1/2) * 8 ft. * 13.86 ft. = 55.44 ft^2.
Since there are four triangular faces, the total surface area of the pyramid is:
Total surface area = area of base + 4 * area of one triangular face = 64 ft^2 + 4 * 55.44 ft^2 = 64 ft^2 + 221.76 ft^2 = 285.76 ft^2.
Therefore, the surface area of the square pyramid is 285.76 ft^2.