Question
Find the surface area of the square pyramid.
Imagine a square pyramid. On one of its slanted sides is a line in the middle connecting to the top point and going down. Near the bottom is a square on the left side, and the line is labelled "4 cm".
Under the pyramid, one of the lines is labelled "6 cm".
A. 64 cm^2
B. 16 cm^2
C. 84 cm^2
D. 48 cm^2
Imagine a square pyramid. On one of its slanted sides is a line in the middle connecting to the top point and going down. Near the bottom is a square on the left side, and the line is labelled "4 cm".
Under the pyramid, one of the lines is labelled "6 cm".
A. 64 cm^2
B. 16 cm^2
C. 84 cm^2
D. 48 cm^2
Answers
Answered by
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 four triangular sides.
### Given:
- The length of one side of the square base is 6 cm.
- The slant height (the line from the midpoint of a side of the base to the apex of the pyramid) is 4 cm.
### Step 1: Calculate the area of the base
The area of the square base is:
\[
\text{Area of base} = \text{side} \times \text{side} = 6 \, \text{cm} \times 6 \, \text{cm} = 36 \, \text{cm}^2
\]
### Step 2: Calculate the area of one triangular face
The area of a triangle is given by:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For each triangular face of the pyramid:
- The base is the same as one side of the square, which is 6 cm.
- The height is the slant height, which is given as 4 cm.
Thus, the area of one triangular face is:
\[
\text{Area of one triangle} = \frac{1}{2} \times 6 \, \text{cm} \times 4 \, \text{cm} = \frac{1}{2} \times 24 \, \text{cm}^2 = 12 \, \text{cm}^2
\]
### Step 3: Calculate the total area of the four triangular faces
Since there are 4 triangular faces:
\[
\text{Total area of triangles} = 4 \times 12 \, \text{cm}^2 = 48 \, \text{cm}^2
\]
### Step 4: Calculate the total surface area of the pyramid
Now, we sum the area of the base and the area of the four triangular faces:
\[
\text{Total surface area} = \text{Area of base} + \text{Total area of triangles}
\]
\[
\text{Total surface area} = 36 \, \text{cm}^2 + 48 \, \text{cm}^2 = 84 \, \text{cm}^2
\]
### Final Answer
The surface area of the square pyramid is:
\[
\boxed{84 \, \text{cm}^2}
\]
Therefore, the answer corresponds to option C: 84 cm².
### Given:
- The length of one side of the square base is 6 cm.
- The slant height (the line from the midpoint of a side of the base to the apex of the pyramid) is 4 cm.
### Step 1: Calculate the area of the base
The area of the square base is:
\[
\text{Area of base} = \text{side} \times \text{side} = 6 \, \text{cm} \times 6 \, \text{cm} = 36 \, \text{cm}^2
\]
### Step 2: Calculate the area of one triangular face
The area of a triangle is given by:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For each triangular face of the pyramid:
- The base is the same as one side of the square, which is 6 cm.
- The height is the slant height, which is given as 4 cm.
Thus, the area of one triangular face is:
\[
\text{Area of one triangle} = \frac{1}{2} \times 6 \, \text{cm} \times 4 \, \text{cm} = \frac{1}{2} \times 24 \, \text{cm}^2 = 12 \, \text{cm}^2
\]
### Step 3: Calculate the total area of the four triangular faces
Since there are 4 triangular faces:
\[
\text{Total area of triangles} = 4 \times 12 \, \text{cm}^2 = 48 \, \text{cm}^2
\]
### Step 4: Calculate the total surface area of the pyramid
Now, we sum the area of the base and the area of the four triangular faces:
\[
\text{Total surface area} = \text{Area of base} + \text{Total area of triangles}
\]
\[
\text{Total surface area} = 36 \, \text{cm}^2 + 48 \, \text{cm}^2 = 84 \, \text{cm}^2
\]
### Final Answer
The surface area of the square pyramid is:
\[
\boxed{84 \, \text{cm}^2}
\]
Therefore, the answer corresponds to option C: 84 cm².