Let the sides of a cube be of length . Then the volume is of length . Now, if the side length is doubled, then becomes . Now substituting that into the volume formula, the new volume is .

In general, increasing a side length by factor will mean that the new side length is and the new volume is therefore .

Or in other words, if the side length of a cube is increased by factor , its volume is increased by factor . This in fact holds for any 3D object (as long as ALL sides are increased by factor ).

Similarly, for any plane (2 dimensional) object, increasing the side length by factor means the area increases by factor (as long as ALL sides are increased by factor ). You can prove this yourself