The question of how shape affects our perception of size has long intrigued psychologists. Young children in particular tend to fixate on the vertical dimension leading them to conclude, for example, that a tall glass holds more water than a short glass regardless of the glass diameter. Although we learn to judge volume better as we age, even adults can be fooled into misjudging relative proportions of different shapes.
Part of the problem stems from the fact that the question of size can be ambiguous in a three-dimensional world. In comparing the lengths of one-dimensional objects, such as bits of string, it's clear that a two-foot long piece is twice as "big" as a one-foot piece, and would weigh twice as much.
For two-dimensional objects, things get a bit trickier. A one-foot tall square has a surface area of one square foot. A two-foot tall square has a surface area of four square feet, four times larger than the small square.
The difference is even more dramatic for three-dimensional objects. A one-foot tall cube has a volume of one cubic foot, but a two-foot tall cube has a volume of eight cubic feet. Although the larger cube is only twice as tall, it is effectively eight times as the size of the smaller cube.
The same is true of more complicated three-dimensional objects, including people. A three-foot tall child might weigh 35 pounds, but a six-foot tall man could weigh 200 pounds or more. Although the man is twice as large, from the perspective of his height, he is nearly six times heavier.
But why is a person different from a cube? That is, why is a six-foot tall man not eight times heavier than a three-foot tall child? Because our bodies change proportions as we grow. An adult is not simply a magnified version of a child, as you can see from the images below. The relationship between height and volume is more complex -- and can be more deceptive.