Pyramid
Pyramids are three dimensional figure like a prism. It is
a solid with polygonal base and triangular lateral faces meeting at a common
point called vertex or apex. This figure has fascinated human beings from the
ancient times. Pyramid of Egypt are one of then seven wonders of the world.
These pyramids were built during the period 3000 - 2000 B.C.
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Pyramids other than tetrahedron are named according to
the name of their bases such as a square pyramid, pentagonal pyramid etc.
Pyramid with triangular base is called tetrahedron.
Pyramid whose base is any regular polygon and other
lateral faces are isosceles triangles is called right regular pyramid otherwise
they are called oblique pyramids.
A cone is also a pyramid, but exceptionally it has a
circular base, not a polygon.
Terms related to a right regular pyramid:
i. Vertex: A common point where the
triangular lateral faces meet is called vertex. In the figure, P is the vertex.
ii. Length
of side of base (a): The
length of the side of base (regular polygon) is called the length of side of
base. In the figure, AB = BC = CD = AD = a is the length of the base.
iii. Vertical
height (h): The
perpendicular distance from vertex to the base is called the vertical height or
simply height of the pyramid. In the figure, OP = h is the height of the
pyramid.
iv. Edge
(e): The
line joining the vertex to the corners of the base is called the edge. In the
figure, PA, PB, PC and PD are the edges. All the edges of a regular pyramid are
equal and it is denoted by e.
v. Slant
height (l): The
line joining the vertex to the mid-point of the side of the base is called the
slant height. It is denoted by l. In the figure, OP is a slant height.
vi. Lateral
faces: The
triangular faces of a pyramid are called lateral faces. In the figure, ΔPAB,
ΔPBC, ΔPCD and ΔPAD are lateral faces. All the lateral faces of a regular
pyramid are congruent triangles.
Relation
among length of base (a), vertical height (h) and slant height (l):
(i) For
square-based pyramid:
(ii) For
equilateral triangle-based pyramid:
Surface area and volume of regular pyramids
(i) For
square-based pyramid:
(ii) For
equilateral triangle-based pyramid:
Workout Examples
Example 1: Find the volume, lateral surface area and total surface
area of the pyramid in which the each side of triangular base is 12 cm and the
slant height is 6 cm.
Solution: Here,
Each
side of base (a) = 12 cm
Slant
height (l) = 6 cm
Thus, Volume = 101.82 cm3,
Lateral surface area = 108 cm2 and Total surface area = 170.35 cm2
Example 2: A pyramid with square base of side 10 cm and height 12 cm
is shown in the figure. Calculate the volume, lateral surface area and the
total surface area of the pyramid.
Solution: Here,
Length
of a side (a) = 10 cm
Vertical
height (l) = 12 cm
Thus, Volume = 400 cm3,
Lateral surface area = 260 cm2 and Total surface area = 360 cm2
You can comment your questions or problems
regarding the surface area and volume of pyramids here.
the upper part of a pyramid has slant height 5 cm and lower part is prism of side 8cm if the volume of solid figure is 448cm, fimd the height of prism.
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