Download Introduction to Prisms A prism is a solid whose ends, or bases, are

Introduction to Prisms
A prism is a solid whose ends, or bases, are parallel congruent polygons, and whose sides or faces
are parallelograms. A straight line moving parallel to itself, its extremities traveling round the outlines of
plane figures, generates the prism.
Prisms are named according to the shape of ends of bases. A prism with a square base is called
a square prism and a prism with hexagonal base is called a hexagonal prism. Similarly, when the ends or
bases of a prism are parallelogram, the prism is called a parallelepiped.
As shown in the figure the sides or faces of the prism are
,
etc. These faces are
parallelograms. If they were rectangles, angle
would be
and the prism would be aright prism. A
prism that is not a right prism is called an oblique prism. A side of one of these parallelograms of these
prisms is called a lateral edge.
The altitude of a prism is the distance between the planes of the two bases (i.e. ). In a right prism,
altitude is the same length as a lateral edge and this is not true for an oblique prism.
Lateral Surface Area of a Prism
The lateral surface area of a prism is the total of the area of the faces.
Lateral Surface Area
Perimeter of the base Height of the prism
Perimeter of the base times the altitude
Rule 1: The lateral area of the prism is equal to the perimeter of the base times the altitude.
Rule 2: Total surface area of a prism is the sum of the lateral areas and the area of its base.
Example:
Find the area of the whole surface of a right triangular prism whose height is
m and the sides of
whose base are
and
m, respectively.
Solution:
In all there are five planes figures, i.e., two triangles and three rectangles. Since both the rectangles
are of equal area.
Area of both triangles
Square meter
Area of all three Rectangles
Square meter
Area of the whole surface
Square meter
Volume of a Prism
Consider the right prism as shown in the figure. There are as many cubic units in each layer parallel
to the base
as there are square units in the area of the base. Also, there are many layers of cubic
units in the prism as there are linear units in the altitudes . We can, therefore, find the total number of
cubic units in the prism by multiplying the area of the base by the altitude.
Volume of the Prism
Area of the Base
Height of the prism
Rule: The volume of the prism equals its base times its altitude.
Example:
The base of a right prism is an equilateral triangle with a side of
its volume.
Solution:
cm and its heights is
cm, find
Volume,
Since, base is an equilateral triangle
cm
cm
Cubic cm
Example:
The sides of a triangular prism are
is
cubic cm. What is its height?
Solution:
Now,
cm,
cm and
cm respectively. The volume of the prism
Square cm
cm
Types of a Prism
Cube:
Let
The cube is a right prism with a square base and a height which is same as the side of the base.
be the side of the cube, then
1. Volume of the cube
area of base
height
i.e.
2. Total surface area of the cube area of six faces
i.e.
3. The line joining the opposite corners of the cube is called the diagonal of the cube. The length of
the diagonal of the cube
Proof: In the given figure, the line
is the diagonal of the cube.
But
Since
and
are the sides of the cube and each has length equal to , therefore
Example:
Three cubes of metal whose edges are in the ratio
diagonal is
Solution:
are melted into a single cube whose
cm. find the edges of three cubes.
Let the edges of the cubes be
Their volumes are:
and
and
cu. cm
and
cu. cm
cm
Volume of the single cube
Let be the edges of the cube then volume,
edge
cu. cm
cm
Now, Diagonal of the cube
cm
But, the diagonal of cube
Hence the three edges of the cube are
Rectangular Prism:
and
(1) Volume of the rectangular prism
(2) Total surface area
area of base
height
area of six faces
(3) Length of the diagonal
(as
,
)
Example:
The length, width and thickness of a rectangular block are
Find the volume, surface area and length of the diagonal of the block.
Solution:
Given that:
cm,
cm,
and
cm
(1) Volume,
cu. cm
(2) Surface area,
sq.cm
(3) Length of diagonal
cm
cm respectively.
Polygonal Prism:
A prism with a polygon base is known as a polygonal prism.
(a) Volume of the prism whose base is a rectangular polygon of
height
sides and height
1.
When sides
is given.
2.
When radius
of inscribed circles is given.
3.
When radius
(b) Lateral surface area
1.
Perimeter of base
When side
of circumscribed circle is given.
height
is given.
2.
When radius
3.
When radius
(c) Total surface area
Lateral surface area
of inscribed circles is given.
of circumscribed circle is given.
Area of base and top
Example:
A pentagonal prism which has its base circumscribed about a circle of radius
height of dm is cast into a cube. Find the size of the cube.
Solution:
Here
dm,
dm,
dm
Since, volume of the material remains the same in both the cases
Volume of the cube Volume of the pentagonal prism
Now, volume of the pentagonal prism
Now by the condition
area of the base
dm, and which has a
or
Taking
both sides, we get
Taking
, we get
dm.