Answer:
L = 315 cm2 ; S = 399.9 cm2
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 triangular pyramid.
Substitute the known value of the base edge length s=14 cm into the formula for the perimeter of the regular triangle P=3s.
P=3(14)=42 cm
Therefore, P=42 cm.
Substitute the known values of the perimeter P=42 cm and the slant height l=15 cm into the formula for the lateral area if a regular pyramid L=12Pl.
L=12(42)(15)=315 cm2
Therefore, the lateral area of the pyramid is 315 cm2.
The surface area of a regular pyramid with lateral area L and base area B is S=L+B, or S=12Pl+B.
The base of the regular triangular pyramid is the equilateral triangle. The area of the triangle with the base b and the height h is B=12bh.
The figure shows an equilateral triangle. The side of the triangle is 14 centimeters long. The angle between the altitude and the side of the triangle measures 30 degrees. Angles between the sides of the triangle measure 60 degrees.
The height is determined by recognizing that it is the long leg of a 30°−60°−90° right triangle. So, the length of the height is 3√2 times the length of the side.
Thus, h=12·3‾√·14
Multiply.
h=73‾√ cm
Therefore, h=73‾√ cm.
Substitute the known values b=14 cm and h=73‾√ cm into the formula for the area of the triangle.
B=12(14)(73‾√)=493‾√ cm2
Therefore, B=493‾√ cm2.
To calculate the surface area of the pyramid, substitute the known values into the formula for surface area. Simplify. Round your answer to the nearest tenth.
S=315+493‾√≈399.9 cm2
Therefore, the surface area of the pyramid is about 399.9 cm2.
