Size matters. But what is size? For a long-jumper or someone planning a car trip, the size that matters is a length, or distance: How far did I jump? How many miles is it from here to there? For a farmer or a real-estate agent, the important measure is area: How many acres is this field? How many square feet of floor space does the house have? Similar issues arise when you go to Home Depot: Molding or pipe is priced by the linear foot, while carpeting is priced by the square foot (area). From a mathematical standpoint the difference is dimensionality. Length or distance is a one-dimensional measurement, while area is two-dimensional.
But we live in a three-dimensional world and that third dimension often matters. We buy milk by the gallon and topsoil by the cubic yard, and our cars have trunk space listed in cubic feet, gas tanks measured in gallons, and engine sizes listed in cubic centimeters. These are all measures of volume. In other situations the important “size” is actually a weight, and for a given material (water, or flour, or aluminum, or beef) the weight is directly proportional to the volume.
In science and engineering, too, all of these measures of “size” are important, depending on the context, and we need to be skilled with all of them. For reasons perhaps known to evolutionary biologists, humans (scientists included) are pretty good at estimating, comparing, and thinking about one-dimensional measures, less adept at working in two dimensions, and largely inept at thinking in three dimensions. (Children seem to be better at three-dimensional reasoning than adults.) So the effort required to master the idea of area is greater than that required for length, and understanding volume requires even more. But that third dimension isn’t going away, and we need to know how to work with it. Just be glad there aren’t four.