If you double the side length of a cube, its surface area becomes 4 times as large and its volume becomes 8 times as large — not simply 2 times! That is because when the similarity ratio is k, area changes by k² and volume by k³. Move the slider to change the similarity ratio and see this relationship for yourself.
Why do we get squares and cubes? Area is made by multiplying two lengths, such as "width × height." If each length becomes k times as long, the area becomes k×k=k² times as large. Volume is made by multiplying three lengths, "width × depth × height," so it becomes k×k×k=k³ times as large.
Real-life example: Compare an ant and an elephant. An elephant may be hundreds of times longer than an ant, but its body weight (which is proportional to volume) can be enormously larger by roughly the cube of that scale. That is why larger animals need legs that are much thicker relative to their bodies to support their weight — the cross-sectional area of their legs grows only by k², while the weight they must support grows by k³.