Question
Find the surface area of a pyramid with a square base of 10yds and a height of 8yds
Answers
GPT-4o mini
To find the surface area of a pyramid with a square base, you need to calculate the area of the base and the area of the four triangular faces.
1. **Calculate the area of the base:**
The base is a square, so its area \( A_{base} \) is given by:
\[
A_{base} = \text{side}^2 = (10 \, \text{yds})^2 = 100 \, \text{yd}^2
\]
2. **Calculate the area of the triangular faces:**
Each triangular face can be calculated as follows. The formula for the area of a triangle is:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
The base of each triangular face is the side length of the square base (10 yards), and we need to find the height of the triangular face.
The height of the triangular face is the slant height of the pyramid. To find the slant height \( l \), we can use the Pythagorean theorem. The height of the pyramid \( h \) is 8 yards, and the half of the base (which is half of 10 yards) is 5 yards. Therefore, we have:
\[
l = \sqrt{h^2 + \left(\frac{\text{base}}{2}\right)^2} = \sqrt{8^2 + 5^2} = \sqrt{64 + 25} = \sqrt{89}
\]
Now we can use the slant height to find the area of one triangular face:
\[
A_{triangle} = \frac{1}{2} \times 10 \, \text{yds} \times \sqrt{89}
\]
To find the total area of the four triangular faces, we multiply by 4:
\[
A_{triangles} = 4 \times \left(\frac{1}{2} \times 10 \, \text{yds} \times \sqrt{89}\right) = 20 \, \text{yds} \times \sqrt{89}
\]
3. **Combine the areas to find the total surface area:**
The total surface area \( A_{total} \) is the area of the base plus the area of the triangular faces:
\[
A_{total} = A_{base} + A_{triangles} = 100 \, \text{yd}^2 + 20 \, \text{yds} \cdot \sqrt{89}
\]
4. **Calculate \( \sqrt{89} \) for a numerical answer:**
\( \sqrt{89} \approx 9.434 \).
Therefore:
\[
A_{triangles} \approx 20 \times 9.434 = 188.68 \, \text{yd}^2
\]
5. **Now calculate the total surface area:**
\[
A_{total} \approx 100 \, \text{yd}^2 + 188.68 \, \text{yd}^2 \approx 288.68 \, \text{yd}^2
\]
Thus, the total surface area of the pyramid is approximately **288.68 square yards**.
1. **Calculate the area of the base:**
The base is a square, so its area \( A_{base} \) is given by:
\[
A_{base} = \text{side}^2 = (10 \, \text{yds})^2 = 100 \, \text{yd}^2
\]
2. **Calculate the area of the triangular faces:**
Each triangular face can be calculated as follows. The formula for the area of a triangle is:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
The base of each triangular face is the side length of the square base (10 yards), and we need to find the height of the triangular face.
The height of the triangular face is the slant height of the pyramid. To find the slant height \( l \), we can use the Pythagorean theorem. The height of the pyramid \( h \) is 8 yards, and the half of the base (which is half of 10 yards) is 5 yards. Therefore, we have:
\[
l = \sqrt{h^2 + \left(\frac{\text{base}}{2}\right)^2} = \sqrt{8^2 + 5^2} = \sqrt{64 + 25} = \sqrt{89}
\]
Now we can use the slant height to find the area of one triangular face:
\[
A_{triangle} = \frac{1}{2} \times 10 \, \text{yds} \times \sqrt{89}
\]
To find the total area of the four triangular faces, we multiply by 4:
\[
A_{triangles} = 4 \times \left(\frac{1}{2} \times 10 \, \text{yds} \times \sqrt{89}\right) = 20 \, \text{yds} \times \sqrt{89}
\]
3. **Combine the areas to find the total surface area:**
The total surface area \( A_{total} \) is the area of the base plus the area of the triangular faces:
\[
A_{total} = A_{base} + A_{triangles} = 100 \, \text{yd}^2 + 20 \, \text{yds} \cdot \sqrt{89}
\]
4. **Calculate \( \sqrt{89} \) for a numerical answer:**
\( \sqrt{89} \approx 9.434 \).
Therefore:
\[
A_{triangles} \approx 20 \times 9.434 = 188.68 \, \text{yd}^2
\]
5. **Now calculate the total surface area:**
\[
A_{total} \approx 100 \, \text{yd}^2 + 188.68 \, \text{yd}^2 \approx 288.68 \, \text{yd}^2
\]
Thus, the total surface area of the pyramid is approximately **288.68 square yards**.