Question

The cloth top of the gazebo in Mr. Chang’s backyard has ripped from weathering. He needs to know the area of the cloth so he can replace it.
A square pyramid with a base of 6 feet by 6 feet. The triangular sides have a height of 5 feet. The height of the pyramid is 4 feet.

How much cloth, in square feet, will Mr. Chang need, assuming he does not need to replace the cloth at the bottom of the figure?

Answer 1

To find the area of the cloth needed to replace the top of the gazebo (the square pyramid), we need to calculate the area of the base and the lateral surface area of the pyramid.

1. **Area of the base**:
The base is a square with side length 6 feet.
\[
\text{Area of the base} = \text{side} \times \text{side} = 6 \, \text{ft} \times 6 \, \text{ft} = 36 \, \text{ft}^2
\]

2. **Lateral surface area**:
The lateral surface area of a square pyramid is given by the formula:
\[
\text{Lateral Surface Area} = \frac{1}{2} \times \text{perimeter of base} \times \text{slant height}
\]

To find the slant height, we can use the Pythagorean theorem. The slant height forms a right triangle with the height of the pyramid and half of the base as follows:
- The half of the base side length is \( \frac{6}{2} = 3 \, \text{ft} \).
- The height of the pyramid is 4 feet.

The slant height \( l \) can be found using the Pythagorean theorem:
\[
l = \sqrt{(\text{height})^2 + \left(\frac{\text{base}}{2}\right)^2} = \sqrt{(4 \, \text{ft})^2 + (3 \, \text{ft})^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \, \text{ft}
\]

Now, we can calculate the perimeter of the base (which is a square):
\[
\text{Perimeter of the base} = 4 \times \text{side} = 4 \times 6 \, \text{ft} = 24 \, \text{ft}
\]

Thus, the lateral surface area is:
\[
\text{Lateral Surface Area} = \frac{1}{2} \times \text{perimeter} \times \text{slant height} = \frac{1}{2} \times 24 \, \text{ft} \times 5 \, \text{ft} = 60 \, \text{ft}^2
\]

3. **Total area of the cloth needed**:
Since Mr. Chang does not need to replace the cloth at the bottom of the figure (the base), we do not include the base area in the total area.
The total area of cloth needed is just the lateral surface area:
\[
\text{Total Area} = \text{Lateral Surface Area} = 60 \, \text{ft}^2
\]

Hence, Mr. Chang needs **60 square feet** of cloth to replace the top of the gazebo.