What is the surface area of a square pyramid where the base edge is 8 ft. and the slant height is 16 ft.?
256 ft.2 squared
320 ft.2 squared
128 ft.2 squared
512 ft.2 squared
The surface area of a square pyramid can be found by adding the area of the base to the sum of the areas of the four triangular faces.
The area of the base can be found by squaring the length of one side (base edge).
Area of base = (base edge)^2 = 8^2 = 64 ft^2
The area of each triangular face can be found using the formula:
Area of triangular face = (1/2) * base * height
Since the triangular faces are isosceles, the base and height of each face are equal. The height of each face can be found using the Pythagorean theorem:
height = √(slant height^2 - (1/2 * base)^2) = √(16^2 - (1/2 * 8)^2) = √(256 - 16) = √240 = 4√15 ft
Therefore, the area of each triangular face is:
Area of triangular face = (1/2) * 8 * 4√15 = 16√15 ft^2
The total surface area is:
Surface area = Area of base + 4 * Area of triangular face
= 64 + 4 * 16√15
= 64 + 64√15
To simplify, we can multiply both terms by √15:
Surface area = 64√15 + 64√(15 * 15)
= 64√15 + 64 * 15
= 64√15 + 960
Since the question asks for the answer in squared units, the final answer would be:
Surface area ≈ 1020 ft^2 squared.
None of the given options match this answer.