Question
The great pyramid of Giza is one of the most famous Egyptian pyramids. The pyramid has a square base and four triangular faces that meet at the top. A side of the base is 230.4 meters and the slant height is 180.3 meters. You are tasked with finding the total surface area of the pyramid, which includes the area of the base and the four triangular faces. round to the nearest hundredths.
Answers
Answered by
GPT-4o mini
To calculate the total surface area of the Great Pyramid of Giza, we need to find 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. The area \(A_{base}\) of the base can be calculated using the formula:
\[
A_{base} = \text{side}^2
\]
Given that the side of the base is 230.4 meters:
\[
A_{base} = (230.4)^2 = 53014.56 \, \text{square meters}
\]
2. **Area of the Triangular Faces:**
Each triangular face (there are four of them) can be calculated using the formula for the area of a triangle:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, the base of each triangular face is the side of the square base (230.4 meters), and the height of the triangle is the slant height (180.3 meters):
\[
A_{triangle} = \frac{1}{2} \times 230.4 \times 180.3
\]
Calculating this gives:
\[
A_{triangle} = \frac{1}{2} \times 230.4 \times 180.3 = \frac{1}{2} \times 41545.92 = 20772.96 \, \text{square meters}
\]
Since there are four triangular faces, the total area of the triangular faces \(A_{triangles}\) is:
\[
A_{triangles} = 4 \times A_{triangle} = 4 \times 20772.96 = 83119.84 \, \text{square meters}
\]
3. **Total Surface Area:**
Finally, 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:
\[
A_{total} = A_{base} + A_{triangles}
\]
Calculating this gives:
\[
A_{total} = 53014.56 + 83119.84 = 136134.40 \, \text{square meters}
\]
Thus, the total surface area of the Great Pyramid of Giza, rounded to the nearest hundredths, is:
\[
\boxed{136134.40} \, \text{square meters}
\]
1. **Area of the Base:**
The base of the pyramid is a square. The area \(A_{base}\) of the base can be calculated using the formula:
\[
A_{base} = \text{side}^2
\]
Given that the side of the base is 230.4 meters:
\[
A_{base} = (230.4)^2 = 53014.56 \, \text{square meters}
\]
2. **Area of the Triangular Faces:**
Each triangular face (there are four of them) can be calculated using the formula for the area of a triangle:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, the base of each triangular face is the side of the square base (230.4 meters), and the height of the triangle is the slant height (180.3 meters):
\[
A_{triangle} = \frac{1}{2} \times 230.4 \times 180.3
\]
Calculating this gives:
\[
A_{triangle} = \frac{1}{2} \times 230.4 \times 180.3 = \frac{1}{2} \times 41545.92 = 20772.96 \, \text{square meters}
\]
Since there are four triangular faces, the total area of the triangular faces \(A_{triangles}\) is:
\[
A_{triangles} = 4 \times A_{triangle} = 4 \times 20772.96 = 83119.84 \, \text{square meters}
\]
3. **Total Surface Area:**
Finally, 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:
\[
A_{total} = A_{base} + A_{triangles}
\]
Calculating this gives:
\[
A_{total} = 53014.56 + 83119.84 = 136134.40 \, \text{square meters}
\]
Thus, the total surface area of the Great Pyramid of Giza, rounded to the nearest hundredths, is:
\[
\boxed{136134.40} \, \text{square meters}
\]