Question
Find the surface area of a square pyramid with side length 4 yd and slant height 4 yd. (4yd, 4yd, 4yd)
Answers
GPT-4o mini
To find the surface area of a square pyramid, we need to calculate the area of the base and the area of the triangular faces.
1. **Area of the base**:
The base of the pyramid is a square, and the area \( A_{base} \) of a square is calculated as:
\[
A_{base} = \text{side length}^2
\]
Here, the side length is 4 yd:
\[
A_{base} = 4 \, \text{yd} \times 4 \, \text{yd} = 16 \, \text{yd}^2
\]
2. **Area of the triangular faces**:
A square pyramid has four triangular faces. The area \( A_{triangle} \) of one triangular face can be calculated using the formula:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
The base of each triangular face is the same as the side length of the square base, which is 4 yd. The height of the triangular face is the slant height of the pyramid, which is also 4 yd:
\[
A_{triangle} = \frac{1}{2} \times 4 \, \text{yd} \times 4 \, \text{yd} = \frac{1}{2} \times 16 \, \text{yd}^2 = 8 \, \text{yd}^2
\]
Since there are four triangular faces, the total area of the triangular faces \( A_{triangles} \) is:
\[
A_{triangles} = 4 \times A_{triangle} = 4 \times 8 \, \text{yd}^2 = 32 \, \text{yd}^2
\]
3. **Total surface area**:
The total surface area \( A_{total} \) of the pyramid is the sum of the area of the base and the area of the triangular faces:
\[
A_{total} = A_{base} + A_{triangles} = 16 \, \text{yd}^2 + 32 \, \text{yd}^2 = 48 \, \text{yd}^2
\]
Therefore, the surface area of the square pyramid is \( \boxed{48 \, \text{yd}^2} \).
1. **Area of the base**:
The base of the pyramid is a square, and the area \( A_{base} \) of a square is calculated as:
\[
A_{base} = \text{side length}^2
\]
Here, the side length is 4 yd:
\[
A_{base} = 4 \, \text{yd} \times 4 \, \text{yd} = 16 \, \text{yd}^2
\]
2. **Area of the triangular faces**:
A square pyramid has four triangular faces. The area \( A_{triangle} \) of one triangular face can be calculated using the formula:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
The base of each triangular face is the same as the side length of the square base, which is 4 yd. The height of the triangular face is the slant height of the pyramid, which is also 4 yd:
\[
A_{triangle} = \frac{1}{2} \times 4 \, \text{yd} \times 4 \, \text{yd} = \frac{1}{2} \times 16 \, \text{yd}^2 = 8 \, \text{yd}^2
\]
Since there are four triangular faces, the total area of the triangular faces \( A_{triangles} \) is:
\[
A_{triangles} = 4 \times A_{triangle} = 4 \times 8 \, \text{yd}^2 = 32 \, \text{yd}^2
\]
3. **Total surface area**:
The total surface area \( A_{total} \) of the pyramid is the sum of the area of the base and the area of the triangular faces:
\[
A_{total} = A_{base} + A_{triangles} = 16 \, \text{yd}^2 + 32 \, \text{yd}^2 = 48 \, \text{yd}^2
\]
Therefore, the surface area of the square pyramid is \( \boxed{48 \, \text{yd}^2} \).