To find the surface area of a rectangular pyramid, you can use the following formula:
\[
\text{Surface Area} = \frac{1}{2} \times (l \times w) + \text{base area} + \text{area of the four triangular sides}
\]
The surface area of a rectangular pyramid includes:
- The area of the base (which is rectangular).
- The area of the four triangular sides.
Given the measurements:
- Length (\(l\)) = 15 m
- Width (\(w\)) = 12 m
- Height (\(h\)) = 10 m
First, we calculate the base area:
\[
\text{Base Area} = l \times w = 15 \times 12 = 180 , \text{m}^2
\]
Next, we need to calculate the area of the triangular sides. The height of the triangular side can be found using the Pythagorean theorem because the height of the pyramid is different from the height of the triangles formed:
The slant height (\(l_s\)) can be calculated for each triangle. The triangles on the length sides have a base of \(w\) and the height equals the slant height, so we need to find the height of the triangle.
For a triangular side, the slant height is calculated using the formula:
\[
l_s = \sqrt{\left( \frac{w}{2} \right)^2 + h^2}
\]
\[
l_s = \sqrt{\left(\frac{12}{2}\right)^2 + 10^2} = \sqrt{6^2 + 10^2} = \sqrt{36 + 100} = \sqrt{136} \approx 11.62 , \text{m}
\]
Now we can calculate the area of the two triangles that have length as their base:
- Area of the two triangular sides with length \(l\):
\[
\text{Area}_{\text{length}} = 2 \times \frac{1}{2} \times l \times l_s = 2 \times \frac{1}{2} \times 15 \times 11.62
\]
Calculating the area:
\[
= 15 \times 11.62 \approx 174.3 , \text{m}^2
\]
- Area of the two triangular sides with width \(w\):
\[
\text{Area}_{\text{width}} = 2 \times \frac{1}{2} \times w \times l_s = 2 \times \frac{1}{2} \times 12 \times 11.62 = 12 \times 11.62 \approx 139.44 , \text{m}^2
\]
Finally, we can calculate the total surface area:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Area}{\text{length}} + \text{Area}{\text{width}}
\]
Substituting the values we calculated:
\[
\text{Total Surface Area} = 180 + 174.3 + 139.44 \approx 493.74 , \text{m}^2
\]
Thus, the surface area of the rectangular pyramid is approximately:
493.74 m².
