Question
Calculate the surface area of the square pyramid that has a base length of 12 and a slant height of 18. Show your work.(2 points)
Write it like a human and show the explanation and equation
Write it like a human and show the explanation and equation
Answers
GPT-4o mini
To calculate the surface area of a square pyramid, we need to find the area of the base and the area of the four triangular sides (lateral surface area).
1. **Calculate the area of the base**:
The base of the pyramid is a square. The area \( A_{\text{base}} \) of the square can be calculated using the formula:
\[
A_{\text{base}} = \text{side}^2
\]
Given that the length of the base (side) is 12, we have:
\[
A_{\text{base}} = 12^2 = 144
\]
2. **Calculate the area of the four triangular faces**:
Each triangular face has a base that is equal to the side length of the square base (which is 12) and a height that corresponds to the slant height of the pyramid. The area \( A_{\text{triangle}} \) of one triangular face can be calculated using the formula:
\[
A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, the base of each triangle is 12 and the height (slant height) is 18. Therefore, the area of one triangular face is:
\[
A_{\text{triangle}} = \frac{1}{2} \times 12 \times 18 = 108
\]
Since there are 4 triangular faces, the total area of the triangular faces is:
\[
A_{\text{triangles}} = 4 \times A_{\text{triangle}} = 4 \times 108 = 432
\]
3. **Calculate the total surface area**:
Now, we can find the total surface area \( A_{\text{total}} \) of the square pyramid by adding the area of the base and the area of the triangular faces:
\[
A_{\text{total}} = A_{\text{base}} + A_{\text{triangles}}
\]
Plugging in the values we calculated:
\[
A_{\text{total}} = 144 + 432 = 576
\]
Therefore, the surface area of the square pyramid is **576 square units**.
1. **Calculate the area of the base**:
The base of the pyramid is a square. The area \( A_{\text{base}} \) of the square can be calculated using the formula:
\[
A_{\text{base}} = \text{side}^2
\]
Given that the length of the base (side) is 12, we have:
\[
A_{\text{base}} = 12^2 = 144
\]
2. **Calculate the area of the four triangular faces**:
Each triangular face has a base that is equal to the side length of the square base (which is 12) and a height that corresponds to the slant height of the pyramid. The area \( A_{\text{triangle}} \) of one triangular face can be calculated using the formula:
\[
A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, the base of each triangle is 12 and the height (slant height) is 18. Therefore, the area of one triangular face is:
\[
A_{\text{triangle}} = \frac{1}{2} \times 12 \times 18 = 108
\]
Since there are 4 triangular faces, the total area of the triangular faces is:
\[
A_{\text{triangles}} = 4 \times A_{\text{triangle}} = 4 \times 108 = 432
\]
3. **Calculate the total surface area**:
Now, we can find the total surface area \( A_{\text{total}} \) of the square pyramid by adding the area of the base and the area of the triangular faces:
\[
A_{\text{total}} = A_{\text{base}} + A_{\text{triangles}}
\]
Plugging in the values we calculated:
\[
A_{\text{total}} = 144 + 432 = 576
\]
Therefore, the surface area of the square pyramid is **576 square units**.