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Calculate the surface area of the square pyramid that has a base length of 12 and a slant height of 18. Show your work.(2 points)
To calculate the surface area of a square pyramid, we need to find the area of the base and the area of the four triangular faces.
1. Area of the Base (a square):
The formula for the area of a square is side length squared. Since the base length is 12, the area of the base is:
Area_base = 12^2 = 144 square units.
2. Area of the Triangular Faces:
The formula for the area of a triangle is 0.5 * base * height. The base of the triangle is the same as the base length of the pyramid, which is 12. The height of each triangular face can be calculated using the Pythagorean theorem:
height = sqrt(slant height^2 - (0.5 * base)^2)
height = sqrt(18^2 - (0.5 * 12)^2)
height = sqrt(324 - 36)
height = sqrt(288) = 12√2
Now, the area of one triangular face is:
Area_triangle = 0.5 * 12 * 12√2 = 72√2 square units.
Since there are four triangular faces on a pyramid, the total area of the triangular faces is:
Total_area_triangles = 4 * 72√2 = 288√2 square units.
3. Total Surface Area:
The total surface area of the square pyramid is the sum of the area of the base and the area of the triangular faces:
Total_surface_area = Area_base + Total_area_triangles
Total_surface_area = 144 + 288√2
Total_surface_area ≈ 588.28 square units
Therefore, the surface area of the square pyramid with a base length of 12 and a slant height of 18 is approximately 588.28 square units.