 ## 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.

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.