Question

Find the surface area of the square pyramid
Height: 4 cm
Base edge: 6 cm
A. 64 cm^2
B. 16 cm^2
C. 84 cm^2
D. 48 cm^2

Answers

GPT-4o mini
To find the surface area of a square pyramid, we need to consider both the base area and the area of the four triangular faces.

1. **Calculate the area of the base**:
The base of the pyramid is a square with each edge measuring 6 cm. The area of the square base \( A_{base} \) is given by:
\[
A_{base} = \text{side}^2 = 6 \, \text{cm} \times 6 \, \text{cm} = 36 \, \text{cm}^2
\]

2. **Calculate the area of the triangular faces**:
Each triangular face has a base equal to the edge of the square (6 cm) and a height that we need to calculate. The height of the triangular face can be found using the Pythagorean theorem.

- First, we find the slant height \( l \) of the pyramid. The slant height is the height of a triangular face. It can be calculated using the height of the pyramid and half the base edge:
\[
\text{Half of base edge} = \frac{6 \, \text{cm}}{2} = 3 \, \text{cm}
\]
Using the Pythagorean theorem:
\[
l = \sqrt{(\text{height})^2 + \left(\frac{\text{base edge}}{2}\right)^2} = \sqrt{(4 \, \text{cm})^2 + (3 \, \text{cm})^2}
\]
\[
l = \sqrt{16 + 9} = \sqrt{25} = 5 \, \text{cm}
\]

3. **Area of one triangular face**:
The area of one triangular face \( A_{triangle} \) is given by:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \, \text{cm} \times 5 \, \text{cm} = 15 \, \text{cm}^2
\]

4. **Total area of the four triangular faces**:
Since there are four triangular faces, we multiply the area of one triangular face by four:
\[
A_{triangles\_total} = 4 \times A_{triangle} = 4 \times 15 \, \text{cm}^2 = 60 \, \text{cm}^2
\]

5. **Total surface area of the pyramid**:
The total surface area \( A_{total} \) is the sum of the base area and the area of the triangular faces:
\[
A_{total} = A_{base} + A_{triangles\_total} = 36 \, \text{cm}^2 + 60 \, \text{cm}^2 = 96 \, \text{cm}^2
\]

Upon checking our calculations, the final surface area is \( 96 \, \text{cm}^2 \). However, since that answer is not among the options provided, it seems there may have been a misinterpretation of the dimensions or a specific requirement such as rounding, estimating, or certain assumptions.

Checking the values again confirms the math holds; if those parameters were supposed to yield values among options A to D, please confirm if the base edge or height has been interpreted correctly. Nevertheless, according to the computed calculations, the surface area stands at 96 cm².

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