##
**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__**:**

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*Workout Examples*

*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 cm*^{3}, Lateral surface area = 108 cm^{2}and Total surface area = 170.35 cm^{2}

*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 cm*^{3}, Lateral surface area = 260 cm^{2}and Total surface area = 360 cm^{2}

*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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