A pyramid is a polyhedron with a polygonal base and triangular faces that meet at a point called an apex which is not in the plane of the polygonal base.
Most of the pyramids that are studied in high school are regular pyramids . These pyramids have the following characteristics:
1 ) The base is a regular polygon. 2 ) All lateral edges are congruent. 3 ) All lateral faces are congruent isosceles triangles. 4 ) The altitude meets the base at its center.
The altitude of a lateral face of a regular pyramid is the slant height. In a non-regular pyramid, slant height is not defined.
Lateral Surface Area
The lateral surface area of a regular pyramid is the sum of the areas of its lateral faces.
The general formula for the lateral surface area of a regular pyramid is L . S . A . = 1 2 p l where p represents the perimeter of the base and l the slant height.
Find the lateral surface area of a regular pyramid with a triangular base if each edge of the base measures 8 inches and the slant height is 5 inches.
The perimeter of the base is the sum of the sides.
p = 3 ( 8 ) = 24 inches
L . S . A . = 1 2 ( 24 ) ( 5 ) = 60 inches 2
Total Surface Area
The total surface area of a regular pyramid is the sum of the areas of its lateral faces and its base. The general formula for the total surface area of a regular pyramid is T . S . A . = 1 2 p l + B where p represents the perimeter of the base, l the slant height and B the area of the base.
Find the total surface area of a regular pyramid with a square base if each edge of the base measures 16 inches, the slant height of a side is 17 inches and the altitude is 15 inches.
The perimeter of the base is 4 s since it is a square.
p = 4 ( 16 ) = 64 inches
The area of the base is s 2 .
B = 16 2 = 256 inches 2
T . S . A . = 1 2 ( 64 ) ( 17 ) + 256 = 544 + 256 = 800 inches 2
There is no formula for a surface area of a non-regular pyramid since slant height is not defined. To find the area, find the area of each face and the area of the base and add them.
The volume of a pyramid equals one-third the area of the base times the altitude (height) of the pyramid. ( V = 1 3 B h ) .
Find the volume of a regular square pyramid with base sides 10 cm and altitude 12 cm.
V = 1 3 B h V = 1 3 ( 10 ) 2 ( 12 ) = 400 cm 2
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