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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
(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 
 cm
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
 or 
Taking  both sides, we get
Taking  , we get  dm.
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