Question
Find the surface area of a square pyramid with side length 3 m and slant height 4 m.
3m, 3 m , 4m,
3m, 3 m , 4m,
Answers
Answered by
GPT-4o mini
To find the surface area of a square pyramid, you need to calculate 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 a side length of \( 3 \, \text{m} \).
\[
\text{Area of the base} = \text{side} \times \text{side} = 3 \, \text{m} \times 3 \, \text{m} = 9 \, \text{m}^2
\]
2. **Area of the Triangular Faces**:
Each triangular face of the pyramid has a base (equal to the side of the square base) and a height equal to the slant height of the pyramid.
The base of each triangular face is \( 3 \, \text{m} \) and the height (slant height) is \( 4 \, \text{m} \).
The area of one triangular face is given by:
\[
\text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \, \text{m} \times 4 \, \text{m} = 6 \, \text{m}^2
\]
Since there are four identical triangular faces, the total area of the triangular faces is:
\[
\text{Total area of triangular faces} = 4 \times 6 \, \text{m}^2 = 24 \, \text{m}^2
\]
3. **Total Surface Area**:
Now, add the area of the base and the total area of the triangular faces to obtain the total surface area.
\[
\text{Total surface area} = \text{Area of the base} + \text{Total area of triangular faces} = 9 \, \text{m}^2 + 24 \, \text{m}^2 = 33 \, \text{m}^2
\]
Therefore, the surface area of the square pyramid is \( \boxed{33 \, \text{m}^2} \).
1. **Area of the Base**:
The base of the pyramid is a square with a side length of \( 3 \, \text{m} \).
\[
\text{Area of the base} = \text{side} \times \text{side} = 3 \, \text{m} \times 3 \, \text{m} = 9 \, \text{m}^2
\]
2. **Area of the Triangular Faces**:
Each triangular face of the pyramid has a base (equal to the side of the square base) and a height equal to the slant height of the pyramid.
The base of each triangular face is \( 3 \, \text{m} \) and the height (slant height) is \( 4 \, \text{m} \).
The area of one triangular face is given by:
\[
\text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \, \text{m} \times 4 \, \text{m} = 6 \, \text{m}^2
\]
Since there are four identical triangular faces, the total area of the triangular faces is:
\[
\text{Total area of triangular faces} = 4 \times 6 \, \text{m}^2 = 24 \, \text{m}^2
\]
3. **Total Surface Area**:
Now, add the area of the base and the total area of the triangular faces to obtain the total surface area.
\[
\text{Total surface area} = \text{Area of the base} + \text{Total area of triangular faces} = 9 \, \text{m}^2 + 24 \, \text{m}^2 = 33 \, \text{m}^2
\]
Therefore, the surface area of the square pyramid is \( \boxed{33 \, \text{m}^2} \).