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

Volume of the cube area of base height i.e.

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 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 three cubes.
Solution:
Let the edges of the cubes be and cm
Their volumes are:
and cu. cm
and cu. cm
Volume of the single cube cu. cm
Let be the edges of the cube then volume,
edge cm
Now, Diagonal of the cube
cm
But, the diagonal of cube
Hence the three edges of the cube are and
Rectangular Prism:
(1) Volume of the rectangular prism area of base height
(2) Total surface area area of six faces
Total surface area
(3) Length of the diagonal
(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) Volume of the prism whose base is a rectangular polygon of sides and height area of the base height

When sides is given.

When radius of inscribed circles is given.

When radius of circumscribed circle is given.
(b) Lateral surface area Perimeter of base height

When side is given.

When radius of inscribed circles is given.

When radius of circumscribed circle is given.
(c) 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 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
Taking both sides, we get
Taking , we get dm.