Question
Find the surface area of the regular pyramid shown to the nearest whole number. The figure is not drawn to scale.
A regular hexagonal pyramid is shown. The slant height is 11 meters, the length of one side of the base is 12 meters, and the apothem of the base is 6 square root 3 meters.
(1 point)
Responses
1,540 m2
1,540 m 2
770 m2
770 m 2
396 m2
396 m 2
749 m2
749 m 2
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To find the surface area of a regular pyramid, we need to calculate the area of the base and the lateral area.
The area of the base can be found using the formula for the area of a regular hexagon. Since the length of one side of the base is 12 meters, the area of the base is:
Area_base = (3√3 / 2) * (side_length)^2
Area_base = (3√3 / 2) * (12)^2
Area_base = (3√3 / 2) * 144
Area_base = (3 * 1.732 / 2) * 144
Area_base = 259.8 square meters
The lateral area can be found by calculating the area of each triangular face and summing them up. Since the slant height is 11 meters and the apothem of the base is 6√3 meters, we can use the formula for the area of a triangle:
Area_triangle = (1/2) * base * height
Area_triangle = (1/2) * 12 * 11
Area_triangle = 66 square meters
There are 6 triangular faces in a regular hexagonal pyramid, so the total lateral area is:
Lateral_area = 6 * Area_triangle
Lateral_area = 6 * 66
Lateral_area = 396 square meters
The surface area is the sum of the base area and the lateral area:
Surface_area = Area_base + Lateral_area
Surface_area = 259.8 + 396
Surface_area = 655.8 square meters
Rounding to the nearest whole number, the surface area of the regular pyramid is approximately 656 square meters. So the correct answer is 656 m2.