L = 6,300 m2 ; S = 15,653.1 m2
Step-by-step explanation:
The lateral area of a regular pyramid with perimeter P and slant height l is L=12Pl.
The figure shows a regular hexagonal pyramid.
Substitute the known value of the base edge length s=60 m into the formula for the perimeter of the base of the regular hexagonal pyramid P=6s.
P=6(60)=360
Therefore, the perimeter of the base of the regular hexagonal pyramid is 360 m.
Substitute the values of the perimeter P=360 m and the slant height l=35 m into the formula for the lateral area of a pyramid, L=12Pl.
L=12(360)(35)=6,300
Therefore, the lateral area of the pyramid is 6,300 m2.
The surface area of a regular pyramid with lateral area L and base area B is S=L+B, or S=12P+B.
As the pyramid is the regular hexagonal pyramid, the base is a regular hexagon.
To find the area of the hexagon, calculate the length of the apothem.
The figure shows a regular hexagon which is divided into 6 equilateral triangles by diagonals. Two vertices coincide with the adjacent vertices of the hexagon. The third vertex is a center of the hexagon. Angles between the diagonals and the sides of the hexagon measure 60 degrees. The apothem from the base of the triangle divides it into two right triangles. The angle between the apothem and the side of the triangle measures 30 degrees. One of the sides of the triangle is 30 meters long. This side is opposite to the angle that measures 30 degrees.
The interior angles of the hexagon have a measure of 120°, so the angles at the base of the triangle shown (and the third angle) have measure 60°.
The height forms a 30°−60°−90° triangle, which has one leg equal to half of the hexagon side, or 30 m.
So, by the 30°−60°−90° Triangle Theorem, a=303‾√ m.
The figure shows the same hexagon as in the previous figure. The apothem is 30 times square root of 3 meters long.
To find the area of the hexagon, use the formula for the area of a regular polygon, B=12aP.
Substitute 303‾√ for a and 360 for P.
B=12·303‾√·360=5,4003‾√
The area of the base is 5,4003‾√ m2.
Use a calculator to approximate. Round your answer to the nearest tenth.
B≈9353.1
To calculate the surface area of the pyramid, substitute the known values into the formula for surface area.
S=6,300+9,353.1=15,653.1
Therefore, the surface area of the pyramid is 15,653.1 m2.
