Question

A solid cube of side $12cm$ is cut into eight cubes of equal volume. What will be the side of the new cube? Also, find the ratio between their surface areas.

Answer 1

Step I: Finding side of the new cube

Side of the big solid cube $=12cm$

We know volume of a cube $=Side\times Side\times Side$

So, Volume of the solid cube $=12\times 12\times 12=1728c{m}^3$

Given that the solid cube is cut into $8$ small cubes of equal volumes.

Thus, Volume of each small cube $=\frac{1}{8}\times 1728=216c{m}^3$

Let the side of small cube is $a.$

So the volume of a small cube $=a\times a\times a=a^3$

Thus, equating the volumes of the small cube, we get:

$a^3=216$

$a=6cm$

Thus, side of the new cube is $6cm$.

Step II: Finding the ratio of surface areas.

We know, the Surface area of cube is given by $SA=6\times sid{e}^2$

Surface area of the big solid cube of side $12cm=6\times {\left(12\right)}^2=864c{m}^2$

Surface area of the small cube of side $6cm=6\times {\left(6\right)}^2=216c{m}^2$

Ratio of $SA$of the cubes $=$ $\frac{SAofsolidcube}{SAofsmallcube}$

$$\begin{gathered} =\frac{864}{216} \\ =\frac{4}{1} \\ =4\mathbf{:}\mathbf{}\mathbf{1} \end{gathered}$$

Thus, the ratio of surface areas of two cubes is $4\mathbf{:}\mathbf{1}$.