The roof of a gazebo is in the shape of an octagonal pyramid. Each side of the octagon is 12 feet, the height of each triangular face is 10 feet, and the area of the octagon is 695 square feet. How many square feet of cedar are needed to cover the lateral surface area of the octagonal pyramid?(1 point) Responses 1,175 ft.2 1,175 ft. squared 360 ft.2 360 ft. squared 480 ft.2 480 ft. squared 960 ft.2
To find the lateral surface area of the octagonal pyramid, we first need to find the slant height of each triangular face using the Pythagorean theorem.
Each triangular face is a right triangle with one leg being the height of the pyramid (10 feet) and the other leg being half the length of one side of the octagon (6 feet).
Using the Pythagorean theorem:
slant height = √(10^2 + 6^2)
slant height = √(100 + 36)
slant height = √136
slant height ≈ 11.66 feet
The lateral surface area of an octagonal pyramid can be found by multiplying the perimeter of the octagon by the slant height and dividing by 2.
Perimeter of octagon = 8 * 12 = 96 feet
Lateral surface area = (96 * 11.66) / 2 = 1121.6 ft^2
Therefore, approximately 1,122 square feet of cedar are needed to cover the lateral surface area of the octagonal pyramid.