Question

Find the total surface area of the cube whose volume is $ 3\sqrt 3 \,{a^3} $ cubic units.

Answer 1

Hint: Here, in this problem to find the total surface area of a cube we first find side of a cube using a formula of volume and then using this value of the side of a cube in the formula of total surface area of a cube to get the required solution of the given problem.
Volume of a cube = $ {\left( {side} \right)^3} $ ,
Total surface area of a cube = $ 6{\left( {side} \right)^2} $

Complete step-by-step answer:
Let side of a cube be x

Then volume of is given as $ {\left( {side} \right)^3} $
 $ \Rightarrow Volume(V) = {\left( {side} \right)^3} $
But, volume of cube is = $ 3\sqrt 3 \,{a^3} $
Substituting value in above formula we have
 $
  3\sqrt 3 {a^3} = {\left( x \right)^3} \\
   \Rightarrow {\left( {\sqrt 3 a} \right)^3} = {\left( x \right)^3} \\
   \Rightarrow x = \sqrt 3 \,a \;
  $
Therefore, side of a cube is $ \sqrt 3 \,\,a\,\,units $
Also, we know that total surface area of a cube is given as $ 6 \times {\left( {side} \right)^2} $
Substituting value of ‘x’ calculated above in above formulas of total surface area. We have,
 $
  Total\,\,surface\,\,area\,\,of\,\,cube = 6 \times {\left( {\sqrt 3 \,\,a} \right)^2} \\
   \Rightarrow Total\,\,surface\,\,area\,\,of\,\,cube = 6 \times \left( {3{a^2}} \right) \\
   \Rightarrow Total\,\,surface\,\,area\,\,of\,\,cube = 18{a^2} \;
  $
Hence, from above we see that the total surface area of the cube is $ 18{a^2} $ .

Note: In mensuration there are different formulas for different figures even there are different formulas for surface area and volumes of figures. So, to get the right solution of the problem students should select appropriate formulas and also do calculations very carefully to get the required solution of the problem.