Question
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
960 ft.2
960 ft. squared
480 ft.2
480 ft. squared
360 ft.2
360 ft. squared
1,175 ft.2
Responses
960 ft.2
960 ft. squared
480 ft.2
480 ft. squared
360 ft.2
360 ft. squared
1,175 ft.2
Answers
GPT-4o mini
To find the lateral surface area of an octagonal pyramid, we first need to calculate the area of the triangular faces.
The lateral surface area of the pyramid consists of 8 triangular faces since the base is an octagon.
The area of one triangular face can be calculated with the formula for the area of a triangle:
\[
\text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For our triangular faces, the base is the side length of the octagon (12 feet), and the height of the triangular face is 10 feet. Thus, the area of one triangular face is:
\[
\text{Area of one triangular face} = \frac{1}{2} \times 12 \, \text{ft} \times 10 \, \text{ft} = \frac{1}{2} \times 120 \, \text{ft}^2 = 60 \, \text{ft}^2
\]
Now, since there are 8 such triangular faces, the total lateral surface area (LSA) of the octagonal pyramid is:
\[
\text{Lateral Surface Area} = 8 \times \text{Area of one triangular face} = 8 \times 60 \, \text{ft}^2 = 480 \, \text{ft}^2
\]
Therefore, the number of square feet of cedar needed to cover the lateral surface area of the octagonal pyramid is **480 ft²**.
The lateral surface area of the pyramid consists of 8 triangular faces since the base is an octagon.
The area of one triangular face can be calculated with the formula for the area of a triangle:
\[
\text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For our triangular faces, the base is the side length of the octagon (12 feet), and the height of the triangular face is 10 feet. Thus, the area of one triangular face is:
\[
\text{Area of one triangular face} = \frac{1}{2} \times 12 \, \text{ft} \times 10 \, \text{ft} = \frac{1}{2} \times 120 \, \text{ft}^2 = 60 \, \text{ft}^2
\]
Now, since there are 8 such triangular faces, the total lateral surface area (LSA) of the octagonal pyramid is:
\[
\text{Lateral Surface Area} = 8 \times \text{Area of one triangular face} = 8 \times 60 \, \text{ft}^2 = 480 \, \text{ft}^2
\]
Therefore, the number of square feet of cedar needed to cover the lateral surface area of the octagonal pyramid is **480 ft²**.