Question

Find the volume \[({{m}^{3}})\] in a cube, one face of which has an area of \[64{{m}^{2}}\].

Answer 1

Hint: First of all, we should use the side of the cube as a. We should know the formulae of one face of the cube and the formulae for volume of cube. We should find the area of one face of the cube and volume of the cube in terms of a. We should equate the area of the cube obtained in terms of a to \[64{{m}^{2}}\]. By this, we have to find the value of a. With this value of a, we have to find the volume of the cube.

Complete step-by-step solution -

Before solving the question, we should know that the length, breadth and height of the cube are equal. We should also know the formula for area of base of cube and formula for volume of cube. We know that the base of the cube is a square. So, let us assume the side of a cube is equal to a. As the base of the cube is a square, the area of the base of the cube of side a is equal to the area of square of side a. So, the formula for the area of the base of the cube is equal to \[{{a}^{2}}\]. We know that the volume of a 3D figure is equal to the product of the area of base and its height. As the side of the cube is equal to a, the height of the cube is also equal to a. So, the volume of the cube is equal to \[{{a}^{3}}\].
So, it is clear that if side of cube = a
Area of one face of cube \[={{a}^{2}}\]
Volume of cube \[={{a}^{3}}\]
In the question, we are given that the area of one face of the cube is equal to \[64{{m}^{2}}\].
We know that the area of one face of a cube is equal to \[{{a}^{2}}\] if a is the side of the cube.
Hence, we get
\[\begin{align}
  & {{a}^{2}}=64{{m}^{2}} \\
 & \Rightarrow a=8m......(1) \\
\end{align}\]
We know that if a is the side of a cube, then the volume of the cube is equal to \[{{a}^{3}}\].
From equation (1), we get
 \[\begin{align}
  & \Rightarrow {{a}^{3}}={{(8m)}^{3}} \\
 & \Rightarrow {{a}^{3}}=216{{m}^{3}} \\
\end{align}\]
Hence, the volume of the cube is equal to \[216{{m}^{3}}\].

Note: Students may go wrong by equating the area of the cube equal to \[64{{m}^{2}}\]. We know that the area of the cube is equal to \[6{{a}^{2}}\].
Now we will get
\[\begin{align}
  & 6{{a}^{2}}=64 \\
 & \Rightarrow {{a}^{2}}=\dfrac{64}{6} \\
 & \Rightarrow a=\sqrt{\dfrac{64}{6}} \\
 & \Rightarrow a=\dfrac{8}{\sqrt{6}} \\
\end{align}\]
From this we will get the value of a as \[\dfrac{8}{\sqrt{6}}\]. But the value of a is equal to 8 which is obtained in the above solution. So, we should be careful what is given in the question and we have to proceed accordingly.