Similar shapes

In general if two shapes are similar, then all linear measurements are proportional: 

                                              (Length of a side in Y)/ (the length of the corresponding side in X)=R.       (1)

Notice that areas can be approximated as sums of small square areas. Sides of the squares are proportional, too. There areas computed as squares of their sides, so

                                                                             (Surface Area of Y)/ (Surface Area of X)=R^2.     (2)


Similarly The volumes can be approximate as the sum of very small cubes, filling a shape. A volume of a cube is a cube of its side length. Therefore 

                                                                      (Volume of Y)/ (Volume of X)=R^3.   (3)

Now lets get back to the given example. Using equation (3) we have 

R^3 =(Volume of Y)/ (Volume of X)=3V/V=3. 
Hence 
\[R^3=3    \Rightarrow  R=\sqrt[3]{3}.\]

Now using equation (2) we get

                                                 (Surface Area of Y)/ (Surface Area of X)=$(\sqrt[3]{3})^2$.

So 

                                                (Surface Area of Y)=$(\sqrt[3]{3})^2$ (Surface Area of X).

It is not part of the question, but using (1) you can also show that 

                                        (Length of a side in Y)=$\sqrt[3]{3}$ (the length of the corresponding side in X)

Please leave a comment if you need any clarifications.