Find the surface area of a rectangular pyramid with these measurements: l = 8 cm, w = 4 cm, and h = 2 cm. Express your answer as a decimal rounded to the nearest hundredth
To find the surface area of a rectangular pyramid, you need to calculate the area of the base and the area of the four triangular faces.
The base is a rectangle, so its area (\( A_{base} \)) can be calculated as the product of its length (\( l \)) and width (\( w \)).
\[ A_{base} = l \times w \]
\[ A_{base} = 8 \, \text{cm} \times 4 \, \text{cm} \]
\[ A_{base} = 32 \, \text{cm}^2 \]
Then, you need to find the slant height of the four triangular faces. The pyramid has two sets of triangular faces, each set sharing a base edge. We need to find the slant heights for both sets.
For the triangles with base \( l \), the slant height (\( s_l \)) can be found using the Pythagorean theorem. This slant height is the hypotenuse of a right triangle where one leg is the height of the pyramid (\( h \)) and the other leg is half the width (\( w/2 \)).
\[ s_l^2 = h^2 + \left( \frac{w}{2} \right)^2 \]
\[ s_l^2 = 2^2 + \left( \frac{4}{2} \right)^2 \]
\[ s_l^2 = 4 + 2^2 \]
\[ s_l^2 = 4 + 4 \]
\[ s_l^2 = 8 \]
\[ s_l = \sqrt{8} \]
\[ s_l \approx 2.83 \, \text{cm} \] (rounded to two decimal places)
For the triangles with base \( w \), the slant height (\( s_w \)) can be found in a similar manner:
\[ s_w^2 = h^2 + \left( \frac{l}{2} \right)^2 \]
\[ s_w^2 = 2^2 + \left( \frac{8}{2} \right)^2 \]
\[ s_w^2 = 4 + 4^2 \]
\[ s_w^2 = 4 + 16 \]
\[ s_w^2 = 20 \]
\[ s_w = \sqrt{20} \]
\[ s_w \approx 4.47 \, \text{cm} \] (rounded to two decimal places)
Now, calculate the area of the four triangles.
The area of one of the triangles with base \( l \) is:
\[ A_{tri_l} = \frac{1}{2} \times l \times s_l \]
\[ A_{tri_l} = \frac{1}{2} \times 8 \times 2.83 \]
\[ A_{tri_l} \approx 11.32 \, \text{cm}^2 \]
Since there are two such triangles, their total area is:
\[ 2 \times A_{tri_l} \approx 2 \times 11.32 \approx 22.64 \, \text{cm}^2 \]
The area of one of the triangles with base \( w \) is:
\[ A_{tri_w} = \frac{1}{2} \times w \times s_w \]
\[ A_{tri_w} = \frac{1}{2} \times 4 \times 4.47 \]
\[ A_{tri_w} \approx 8.94 \, \text{cm}^2 \]
Since there are two such triangles, their total area is:
\[ 2 \times A_{tri_w} \approx 2 \times 8.94 \approx 17.88 \, \text{cm}^2 \]
The total surface area (\( A_{total} \)) is the sum of the base area and the areas of the four triangular faces.
\[ A_{total} = A_{base} + 2 A_{tri_l} + 2 A_{tri_w} \]
\[ A_{total} \approx 32 + 22.64 + 17.88 \]
\[ A_{total} \approx 72.52 \, \text{cm}^2 \]
Rounded to the nearest hundredth, the surface area of the rectangular pyramid is approximately 72.52 square centimeters.