Question

What is the surface area of a square pyramid where the base edge is 8 ft. And the slant height is 16 ft?

Answer 1

To find the surface area of a square pyramid, we need to calculate the area of each face and then sum them up.

The base of the pyramid is a square, so the area of the base is simply the length of one edge squared: 8 ft * 8 ft = 64 ft^2.

The slant height is the height of each triangular face. In a square pyramid, all four triangular faces are congruent isosceles triangles.

Using the Pythagorean theorem, we can find the height of each triangular face:

height^2 + base length^2 = slant height^2
height^2 + 8 ft^2 = 16 ft^2
height^2 = 16 ft^2 - 8 ft^2
height^2 = 256 ft^2 - 64 ft^2
height^2 = 192 ft^2

Taking the square root of both sides, we find that the height of each triangular face is √192 ft.

The area of each triangular face is given by 0.5 * base length * height = 0.5 * 8 ft * √192 ft = 4 ft * √192 ft.

Since we have four identical triangular faces, the total area of the triangular faces is 4 * 4 ft * √192 ft = 16 ft * √192 ft.

Thus, the surface area of the square pyramid is the sum of the base area and the area of the triangular faces:

Surface area = base area + area of triangular faces
Surface area = 64 ft^2 + 16 ft * √192 ft
Surface area ≈ 64 ft^2 + 146.91 ft^2 (rounded to two decimal places)

Therefore, the surface area of the square pyramid is approximately 210.91 ft^2 (rounded to two decimal places).