Question

A solid cube is cut into 27 identical cubes . What is the percentage increase in the total surface area?

Answer 1

So lets say the side length of the original cube is 'a'.

Now the volume of the original cube is equal to $a^3$

and the surface area of the original cube $=6a^2$

If the cube is cut into 27 identical cubes then the volume of each cube is:

$a_n^3=\frac{a^3}{27}$

$a_n=\frac{a}{3}$

where, $a_n$ is the side length of each smaller cube. So, the surface area of each tiny cube is:

$6a_n^2=6\times (\frac{a}{3} )^2=6\times \frac{a^2}{9} =\frac{6a^2}{9}$

Now, there are 27 identical cubes so the total surface area is:

$27 \times \frac{6a^2}{9} =18a^2$

Now, the surface area of the original cube is $6a^2$

and surface area of 27 identical cubes is $18a^2$

So, the percentage increase in the total surface area is:

Percentage increase $=\frac{N-O}{O}$, where N stands for new value, O stands for the old value.

$\frac{18a^2-6a^2}{6a^2} \times 100=\frac{12a^2}{6a^2} \times 100=2\times 100=200$

So the percentage increase in the total surface area is 200%.