Question
What is the total surface area of the pyramid?
mc021-1.jpg
195.6 square centimeters
217.8 square centimeters
255.6 square centimeters
293.4 square centimeters
mc021-1.jpg
195.6 square centimeters
217.8 square centimeters
255.6 square centimeters
293.4 square centimeters
Answers
GPT-4o mini
To find the total surface area of a pyramid, we need to calculate the area of its base and the areas of its triangular faces.
Let's denote the following:
- \( B \) = area of the base
- \( L \) = slant height of the pyramid
- \( A_f \) = area of one triangular face
- \( n \) = number of triangular faces (for a square pyramid, this is 4)
The total surface area \( S \) of a pyramid can be calculated using the formula:
\[
S = B + n \cdot A_f
\]
1. **Area of the base**: If the base is a square, the area \( B \) would be \( s^2 \), where \( s \) is the length of one side of the square base.
2. **Area of one triangular face**: The area of one triangular face is given by:
\[
A_f = \frac{1}{2} \cdot \text{base} \cdot \text{height}
\]
For a triangular face of a square pyramid, the base would be the length of one side of the square \( s \), and the height would be the slant height \( L \):
\[
A_f = \frac{1}{2} \cdot s \cdot L
\]
3. **Calculate total surface area**:
\[
S = B + n \cdot A_f
\]
For a square base:
\[
S = s^2 + 4 \cdot \left(\frac{1}{2} \cdot s \cdot L\right)
\]
This simplifies to:
\[
S = s^2 + 2sL
\]
To calculate the exact total surface area, we need the dimensions of the base and the slant height from the image you referenced. Since I cannot view the image (mc021-1.jpg), please provide the measurements of the base (side length \( s \)) and the slant height \( L \) if available.
Once we have these measurements, we can perform the calculations and determine which of the options is the correct total surface area.
Let's denote the following:
- \( B \) = area of the base
- \( L \) = slant height of the pyramid
- \( A_f \) = area of one triangular face
- \( n \) = number of triangular faces (for a square pyramid, this is 4)
The total surface area \( S \) of a pyramid can be calculated using the formula:
\[
S = B + n \cdot A_f
\]
1. **Area of the base**: If the base is a square, the area \( B \) would be \( s^2 \), where \( s \) is the length of one side of the square base.
2. **Area of one triangular face**: The area of one triangular face is given by:
\[
A_f = \frac{1}{2} \cdot \text{base} \cdot \text{height}
\]
For a triangular face of a square pyramid, the base would be the length of one side of the square \( s \), and the height would be the slant height \( L \):
\[
A_f = \frac{1}{2} \cdot s \cdot L
\]
3. **Calculate total surface area**:
\[
S = B + n \cdot A_f
\]
For a square base:
\[
S = s^2 + 4 \cdot \left(\frac{1}{2} \cdot s \cdot L\right)
\]
This simplifies to:
\[
S = s^2 + 2sL
\]
To calculate the exact total surface area, we need the dimensions of the base and the slant height from the image you referenced. Since I cannot view the image (mc021-1.jpg), please provide the measurements of the base (side length \( s \)) and the slant height \( L \) if available.
Once we have these measurements, we can perform the calculations and determine which of the options is the correct total surface area.