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.

Question

____m^2

Answer 1

To find the surface area of a rectangular pyramid, we need to calculate both the area of the base and the areas of the four triangular faces.

1. Calculate the area of the base: The base is a rectangle, so its area $A_{base}$ can be calculated as: $$A_{base}=l\times w=15,\text{m}\times 12,\text{m}=180,{\text{m}}^2$$

2. Calculate the area of the triangular faces: Each triangular face has a base that corresponds to one side of the rectangle and a height that extends from the apex of the pyramid to the midpoint of that base.

   The height of each triangular face can be found using the Pythagorean theorem. The slant height $s$ can be calculated from the pyramid height $h$ and half of the base length of the triangle being analyzed.

   • For the triangles with bases $l=15,\text{m}$:

     • Half of the base = $\frac{l}{2}=\frac{15}{2}=7.5,\text{m}$
     • The slant height $s$ can be calculated as: $$s=\sqrt{h^2+{\left(\frac{l}{2}\right)}^2}=\sqrt{{10}^2+{7.5}^2}=\sqrt{100+56.25}=\sqrt{156.25}\approx 12.5,\text{m}$$
     • Area of each triangular face with base $l$: $$A_{triangle1}=\frac{1}{2}\times base\times height=\frac{1}{2}\times 15,\text{m}\times 12.5,\text{m}=\frac{1}{2}\times 15\times 12.5=93.75,{\text{m}}^2$$
     • Since there are 2 triangular faces with this base: $$A_{triangle{s}_l}=2\times 93.75=187.5,{\text{m}}^2$$

   • For the triangles with bases $w=12,\text{m}$:

     • Half of the base = $\frac{w}{2}=\frac{12}{2}=6,\text{m}$
     • Calculate the slant height: $$s=\sqrt{h^2+{\left(\frac{w}{2}\right)}^2}=\sqrt{{10}^2+6^2}=\sqrt{100+36}=\sqrt{136}\approx 11.66,\text{m}$$
     • Area of each triangular face with base $w$: $$A_{triangle2}=\frac{1}{2}\times base\times height=\frac{1}{2}\times 12,\text{m}\times 11.66,\text{m}=\frac{1}{2}\times 12\times 11.66=69.96,{\text{m}}^2$$
     • Since there are also 2 triangular faces with this base: $$A_{triangle{s}_w}=2\times 69.96=139.92,{\text{m}}^2$$

3. Total surface area: Combine the areas: $$A_{total}=A_{base}+A_{triangle{s}_l}+A_{triangle{s}_w}=180,{\text{m}}^2+187.5,{\text{m}}^2+139.92,{\text{m}}^2=507.42,{\text{m}}^2$$

Thus, the surface area of the rectangular pyramid is approximately: $$\boxed{{\displaystyle 507.42,{\text{m}}^2}}$$