Question

: If the volume of a cube is $3\sqrt 3 {a^3}$ units, then find its total surface area.

Answer 1

Hint: First, the volume of the cube means its total space will only need to be occupied by the cube.
It is the capacity of the given cube, and the amount of water that will be contained in the ice cubes is equal to the volume of the cube in ice.
Hence from this volume of the cube is given $3\sqrt 3 {a^3}$ units, we need to find its total surface area.
Formula used: Total surface of the area of a cube is $6{x^2}$

Complete step by step answer:
The general volume of the given cube’s formula has been generalized as any solid object is the multiplication of its original length, breadth, and height.
From this, the cube is the same so the volume of the cube is taken as the cube length of its sides.
Hence volume = length $ \times $ breadth $ \times $ height and the sides of the volume is x.
Therefore, we get the volume of the formula is $x \times x \times x = {x^3}$
Given that, If the volume of a cube is $3\sqrt 3 {a^3}$ units, then apply this in the formula we found.
Thus, we get, ${x^3} = (3\sqrt 3 {a^3}) \Rightarrow {(\sqrt 3 a)^3}$ (taking the cube root in common as ${(\sqrt 3 a)^3} = (3\sqrt 3 {a^3})$)
Hence canceling each other we get, ${x^3} = {(\sqrt 3 a)^3} \Rightarrow x = \sqrt 3 a$
Now we are going to find the total surface of the given volume of the cube, which is $6{x^2}$
Substituting the value of x in the total surface formula we get, $6{x^2} = 6{(\sqrt 3 a)^2} \Rightarrow 6 \times 3{a^2}$
Further solving this we get, $18{a^2}$ is the total surface of the given volume cube.

Note: Total surface is the sum of the given areas of all faces of the cube, it will cover all the faces.
There are six times the given square length of the sides of the cube is equivalent, hence it will be represented as $6{x^2}$ where the given x is the length of the sides of the cube.
By the use of this information, we calculated the total surface of the given volume of the cube.