To find the surface area of a rectangular pyramid, you need to know the dimensions of the base and the height of the pyramid. The values you provided (11 in, 11 in, 12 in, and 12 in) suggest we may be dealing with a rectangular base, where the length and width have values of 11 inches and 12 inches, respectively.
Let's calculate the surface area step by step:
- Base Area: The base is a rectangle with a length of 12 inches and a width of 11 inches.
\[ \text{Base Area} = \text{Length} \times \text{Width} = 12, \text{in} \times 11, \text{in} = 132, \text{in}^2 \]
- Lateral Area: In a rectangular pyramid, the lateral area consists of four triangular faces. The area of each triangular face is given by:
\[ \text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
To calculate the lateral surface area, we need to find the slant height of the pyramid. Since the height (the vertical distance from the apex of the pyramid to the center of the base) isn't provided directly, we would require that information, or the problem might imply symmetry in some way.
Here, I will assume that the height is provided as a side length (12 in is often interpreted as a height in similar problems), with the two triangular faces based on the dimensions of the base:
- Two triangles will have a base of 12 inches and a triangle height that's the slant height (\( l_1 \)).
- The other two triangles will have a base of 11 inches and a triangle height that's the slant height (\( l_2 \)).
If we assume the height of the pyramid from the apex to the center of the base is h, we need that value to proceed further with the calculation of the slant heights.
Surface Area Calculation:
Assuming:
- \( h = 12\) in
- Calculate the slant height using the Pythagorean theorem.
For the triangles with a base of 12 in: \[ l_1 = \sqrt{\left(\frac{12}{2}\right)^2 + h^2} = \sqrt{(6)^2 + (12)^2} = \sqrt{36 + 144} = \sqrt{180} = 6\sqrt{5} \approx 13.42 \text{ in} \]
For the triangles with a base of 11 in: \[ l_2 = \sqrt{\left(\frac{11}{2}\right)^2 + h^2} = \sqrt{(5.5)^2 + (12)^2} = \sqrt{30.25 + 144} = \sqrt{174.25} \approx 13.19 \text{ in} \]
Now, we can calculate the lateral area:
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Area of 2 triangles with base 12 in: \[ \text{Area} = 2 \times \left(\frac{1}{2} \times 12 \times l_1\right) = 12 \times l_1 = 12 \times 13.42 \approx 161.04 \text{ in}^2 \]
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Area of 2 triangles with base 11 in: \[ \text{Area} = 2 \times \left(\frac{1}{2} \times 11 \times l_2\right) = 11 \times l_2 = 11 \times 13.19 \approx 145.09 \text{ in}^2 \]
Combining the areas: \[ \text{Lateral Area} = 161.04 + 145.09 \approx 306.13 \text{ in}^2 \]
- Total Surface Area: \[ \text{Total Surface Area} = \text{Base Area} + \text{Lateral Area} = 132 + 306.13 \approx 438.13 \text{ in}^2 \]
Keep in mind that without a clear specification of the height, the stated assumptions are varied, and it would be useful to clarify that value for a more exact computation of surface area.
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