Question

If S is the total surface area of a cube and V is its volume, then which one of the following is correct?
A.${V^3} = 216{S^2}$
B.${S^3} = 216{V^2}$
C.${S^3} = 6{V^2}$
D.${S^2} = 36{V^3}$

Answer 1

Hint: First we need to write the volume and the total surface area of a cube and then determine the relation between them.

Complete step-by-step answer:

Let a cube be there with each edge measuring $a{\text{ cm}}$.
First, we need to find the volume of the cube.
We know that the area of the square with side $a$ is equal to ${a^2}{\text{ }}c{m^2}$.
This square forms the face of a cube.
Then for cube, Volume is given by
V = Area of face of cube x Height of cube
$ \Rightarrow {\text{V = }}{a^2} \times a = {a^3}{\text{ }}c{m^3}{\text{ (1)}}$
Now we need to find the surface area of the cube.
As we already mentioned that area of each face of a cube ${a^2}{\text{ }}c{m^2}$.
Also, a cube consists of 6 of these faces.
Hence Surface area of cube will be given by
${\text{S}} = 6{a^2}{\text{ }}c{m^2}{\text{ (2)}}$
Since we have obtained the formula of volume and surface area of a cube, we can obtain the relation between the same.
From equations $(1)$ and $(2)$, we get
$
  {\text{V}} = {a^3} \\
  {\text{S}} = 6{a^2} \\
$
Cubing equation $(2)$, we get
$
  {{\text{S}}^3} = 216{a^6} = 216{\left( {{a^3}} \right)^2} \\
   \Rightarrow {{\text{S}}^3} = 216{{\text{V}}^2} \\
$
Hence the relation between volume and surface area of a cube is ${{\text{S}}^3} = 216{{\text{V}}^2}$.
Therefore, (B) ${{\text{S}}^3} = 216{{\text{V}}^2}$ is the correct answer.

Note: The volume and the surface area of a cube can be calculated as explained in the above solution and should be kept in mind for solving problems like above. The above relation is only valid for cube.