Prism Types
Cube
The cube is a right prism with a square base and a height which is same as the side of the base. Let be the side of the cube, then

The volume of the cube area of base height i.e.

The total surface area of the cube area of six faces i.e.

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 a length equal to , therefore
Example:
Three cubes of metal whose edges are in the ratio are melted into a single cube whose diagonal is cm. Find the edges of the three cubes.
Solution:
Let the edges of the cubes be and cm
their volumes are:
and cu. cm
and cu. cm
the volume of the single cube cu. cm
Let be the edges of the cube, then volume:
edge cm
Now, the diagonal of the cube
cm
But, the diagonal of the cube
Hence the three edges of the cube are and
Rectangular Prism
(1) The volume of the rectangular prism area of base height
(2) The total surface area area of six faces
The total surface area
(3) The length of the diagonal is
(as, )
Example:
The length, width and thickness of a rectangular block are and cm respectively. Find the volume, surface area and length of the diagonal of the block.
Solution:
Given that: cm, cm, cm
(1) Volume cu. cm
(2) Surface area sq.cm
(3) Length of diagonal
Polygonal Prism
A prism with a polygon base is known as a polygonal prism.
(a) The volume of a prism whose base is a rectangular polygon of sides and height area of the base height.

when side is given.

when the radius of the inscribed circle is given.

when the radius of the circumscribed circle is given.
(b) The lateral surface area perimeter of base height

when side is given.

when the radius of the inscribed circle is given.

when the radius of the circumscribed circle is given.
(c) The total surface area lateral surface area area of base and top
Example:
A pentagonal prism which has its base circumscribed about a circle of radius dm, and which has a height of dm is formed into a cube. Find the size of the cube.
Solution:
Here dm, dm, dm
Since the volume of the material remains the same in both cases
the volume of the cube the volume of the pentagonal prism.
Now, the volume of the pentagonal prism is
Now by the condition
Taking both sides, we get
Taking , we get dm.