Question
Find the surface area of a rectangular pyramid with these measurements:
l = 15 m,
w = 12 m, and
h = 10 m. Express your answer as a decimal rounded to the nearest hundredth.
l = 15 m,
w = 12 m, and
h = 10 m. Express your answer as a decimal rounded to the nearest hundredth.
Answers
Answered by
GPT-4o mini
To find the surface area of a rectangular pyramid, we need to calculate the area of the base and the area of the four triangular faces.
1. **Base Area**:
The base is a rectangle, so the area can be calculated using the formula:
\[
\text{Base Area} = l \times w
\]
where \( l = 15 \) m and \( w = 12 \) m.
\[
\text{Base Area} = 15 \times 12 = 180 \text{ m}^2
\]
2. **Area of the Triangular Faces**:
Each triangular face has a base equal to one side of the rectangle and an apex at the height of the pyramid. We need to calculate the slant heights for each triangular face, which we can find using the Pythagorean theorem.
The slant height can be found for the triangular faces on the length side and the width side.
- **For the triangular face with base \( l = 15 \) m**:
\[
\text{Slant height} (l) = \sqrt{\left(\frac{w}{2}\right)^2 + h^2} = \sqrt{\left(\frac{12}{2}\right)^2 + 10^2} = \sqrt{6^2 + 10^2} = \sqrt{36 + 100} = \sqrt{136} = 2\sqrt{34} \approx 11.66 \text{ m}
\]
The area of this triangular face is:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 15 \times 11.66 \approx 87.45 \text{ m}^2
\]
Since there are two of these triangular faces, the total area contribution from these is:
\[
2 \times 87.45 = 174.90 \text{ m}^2
\]
- **For the triangular face with base \( w = 12 \) m**:
\[
\text{Slant height} (w) = \sqrt{\left(\frac{l}{2}\right)^2 + h^2} = \sqrt{\left(\frac{15}{2}\right)^2 + 10^2} = \sqrt{7.5^2 + 10^2} = \sqrt{56.25 + 100} = \sqrt{156.25} = 12.5 \text{ m}
\]
The area of this triangular face is:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 12.5 = 75.00 \text{ m}^2
\]
Since there are also two of these triangular faces, the total area contribution from these is:
\[
2 \times 75.00 = 150.00 \text{ m}^2
\]
3. **Total Surface Area**:
Now, we sum up all the areas:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Area from triangular faces with length} + \text{Area from triangular faces with width}
\]
\[
\text{Total Surface Area} = 180 + 174.90 + 150 = 504.90 \text{ m}^2
\]
Therefore, the surface area of the rectangular pyramid is:
\[
\boxed{504.90} \text{ m}^2
\]
1. **Base Area**:
The base is a rectangle, so the area can be calculated using the formula:
\[
\text{Base Area} = l \times w
\]
where \( l = 15 \) m and \( w = 12 \) m.
\[
\text{Base Area} = 15 \times 12 = 180 \text{ m}^2
\]
2. **Area of the Triangular Faces**:
Each triangular face has a base equal to one side of the rectangle and an apex at the height of the pyramid. We need to calculate the slant heights for each triangular face, which we can find using the Pythagorean theorem.
The slant height can be found for the triangular faces on the length side and the width side.
- **For the triangular face with base \( l = 15 \) m**:
\[
\text{Slant height} (l) = \sqrt{\left(\frac{w}{2}\right)^2 + h^2} = \sqrt{\left(\frac{12}{2}\right)^2 + 10^2} = \sqrt{6^2 + 10^2} = \sqrt{36 + 100} = \sqrt{136} = 2\sqrt{34} \approx 11.66 \text{ m}
\]
The area of this triangular face is:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 15 \times 11.66 \approx 87.45 \text{ m}^2
\]
Since there are two of these triangular faces, the total area contribution from these is:
\[
2 \times 87.45 = 174.90 \text{ m}^2
\]
- **For the triangular face with base \( w = 12 \) m**:
\[
\text{Slant height} (w) = \sqrt{\left(\frac{l}{2}\right)^2 + h^2} = \sqrt{\left(\frac{15}{2}\right)^2 + 10^2} = \sqrt{7.5^2 + 10^2} = \sqrt{56.25 + 100} = \sqrt{156.25} = 12.5 \text{ m}
\]
The area of this triangular face is:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 12.5 = 75.00 \text{ m}^2
\]
Since there are also two of these triangular faces, the total area contribution from these is:
\[
2 \times 75.00 = 150.00 \text{ m}^2
\]
3. **Total Surface Area**:
Now, we sum up all the areas:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Area from triangular faces with length} + \text{Area from triangular faces with width}
\]
\[
\text{Total Surface Area} = 180 + 174.90 + 150 = 504.90 \text{ m}^2
\]
Therefore, the surface area of the rectangular pyramid is:
\[
\boxed{504.90} \text{ m}^2
\]