Find the surface area of the regular pyramid shown to the nearest whole number.
A picture of a six-sided pyramid with height 16 m, side is 10 m and width from center point of base pentagon to side is start root 3 end root. Label: Not drawn to scale.
(1 point)
Responses
740 m2
740 m 2
480 m2
480 m 2
1,480 m2
1,480 m 2
715 m2
To find the surface area of a regular pyramid, we need to calculate the area of the base and the lateral surface area.
First, let's find the area of the base.
The base is a regular pentagon with side length 10 m.
Since a regular pentagon can be divided into five congruent triangles, we can calculate the area of one of these triangles.
The apothem of the base is given by a = √3, and the side length is given by s = 10 m.
The area of one triangular face is (1/2) * base * height = (1/2) * 10 m * (√3) m = 5√3 m^2.
Since there are five triangular faces making up the base, the total area of the base is 5 * 5√3 m^2 = 25√3 m^2.
Now, let's find the lateral surface area.
The lateral surface area of a regular pyramid can be calculated using the formula:
Lateral surface area = (1/2) * perimeter of base * slant height.
The perimeter of the base pentagon is 10 m * 5 = 50m.
The slant height is the height of one of the triangular faces, which can be found using Pythagorean theorem:
h = √(16 m^2 + (5/2)√3 m)^2 = √(256 m^2 + 75 m^2) = √331 m.
Therefore, the lateral surface area = (1/2) * 50 m * √331 m = 25√331 m^2.
Finally, to find the total surface area, we add the area of the base and the lateral surface area:
Total surface area = 25√3 m^2 + 25√331 m^2 = 25(√3 + √331) m^2.
Approximating this value to the nearest whole number, the surface area is approximately 480 m^2.
Therefore, the correct answer is:
480 m2.
480 m 2.