Question
For the following question, find the surface area of the regular pyramid shown to the nearest whole number.
A hexagonal pyramid is shown. A red segment extends from the center of the hexagonal base to a side. A small square is located at the intersection of this segment with the side. The measure of the segment is 5 start root 3 end root. The length of a side of the hexagonal base is 10 meters. The slant height is 13 meters.
(1 point)
Responses
1,300 m2
1,300 m 2
390 m2
390 m 2
650 m2
650 m 2
628 m2
628 m 2
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To find the surface area of the regular pyramid, we need to find the area of the base and the lateral faces.
The base of the pyramid is a regular hexagon, and the formula to find the area of a regular hexagon is A = (3√3/2) * s^2, where s is the length of a side. In this case, s is given as 10 meters.
Therefore, the area of the base (A_base) is A_base = (3√3/2) * (10^2) = (3√3/2) * 100 = (300√3/2) = 150√3 square meters.
Now, let's find the area of each lateral face. The lateral face is a triangle with the base as the side of the hexagon and the height as the slant height. The formula to find the area of a triangle is A = (1/2) * b * h.
Plugging in the values, A_lateral = (1/2) * 10 * 13 = 65 square meters.
Since there are 6 identical lateral faces in a hexagonal pyramid, the total area of the lateral faces (A_lateral_total) is 6 * 65 = 390 square meters.
Finally, the total surface area is given by A_total = A_base + A_lateral_total = 150√3 + 390 = 150√3 + 390 ≈ 540.63 + 390 ≈ 930.63.
Therefore, the surface area of the regular pyramid, to the nearest whole number, is 931 square meters.
The correct answer is (D) 931 m2.