Question
Find the surface area of a square pyramid with side length 6 m and slant height 7 m.
6m, 6m, 7m
6m, 6m, 7m
Answers
GPT-4o mini
To calculate the surface area of a square pyramid, we need to consider both the area of the base and the area of the four triangular faces.
1. **Area of the base**: The base of the pyramid is a square with side length \( s = 6 \, \text{m} \).
\[
\text{Area of the base} = s^2 = 6^2 = 36 \, \text{m}^2
\]
2. **Area of the triangular faces**: Each triangular face has a base equal to the side length of the pyramid and a height equal to the slant height. The area \( A \) of one triangular face can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is \( 6 \, \text{m} \) and the height (slant height) is \( 7 \, \text{m} \):
\[
A = \frac{1}{2} \times 6 \times 7 = \frac{1}{2} \times 42 = 21 \, \text{m}^2
\]
Since there are 4 triangular faces, the total area of the triangular faces is:
\[
\text{Total area of triangular faces} = 4 \times 21 = 84 \, \text{m}^2
\]
3. **Total surface area**: Now, we can find the total surface area \( S \) of the pyramid by adding the area of the base and the area of the triangular faces:
\[
S = \text{Area of the base} + \text{Total area of triangular faces} = 36 \, \text{m}^2 + 84 \, \text{m}^2 = 120 \, \text{m}^2
\]
Thus, the total surface area of the square pyramid is **120 m²**.
1. **Area of the base**: The base of the pyramid is a square with side length \( s = 6 \, \text{m} \).
\[
\text{Area of the base} = s^2 = 6^2 = 36 \, \text{m}^2
\]
2. **Area of the triangular faces**: Each triangular face has a base equal to the side length of the pyramid and a height equal to the slant height. The area \( A \) of one triangular face can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is \( 6 \, \text{m} \) and the height (slant height) is \( 7 \, \text{m} \):
\[
A = \frac{1}{2} \times 6 \times 7 = \frac{1}{2} \times 42 = 21 \, \text{m}^2
\]
Since there are 4 triangular faces, the total area of the triangular faces is:
\[
\text{Total area of triangular faces} = 4 \times 21 = 84 \, \text{m}^2
\]
3. **Total surface area**: Now, we can find the total surface area \( S \) of the pyramid by adding the area of the base and the area of the triangular faces:
\[
S = \text{Area of the base} + \text{Total area of triangular faces} = 36 \, \text{m}^2 + 84 \, \text{m}^2 = 120 \, \text{m}^2
\]
Thus, the total surface area of the square pyramid is **120 m²**.