Measurements come in three flavours and only one of them behaves the way intuition expects. Lengths scale directly. Areas scale with the square of the length. Volumes scale with the cube.
length × k → area × k² → volume × k³
Everything in this post is that one line, applied to places where it costs money.
Round things are sold on the wrong number
A 9 inch pizza has 63.6 square inches of surface. A 13 inch pizza has 132.7. The pizza value calculator puts the larger one at 108.6 per cent bigger, for a diameter only 44 per cent greater.
At $14 and $22 the small one gives 4.54 square inches per dollar and the large one 6.03, so the large is a third better value despite costing more than half as much again. The calculator also runs the comparison people actually face at the counter: two 9 inch pizzas cost $28 and give 127.2 square inches, while one 13 inch gives 132.7 for $22.
The same geometry runs through screens, where the advertised number is the diagonal. Going from a 55 inch television to a 65 inch sounds like an 18 per cent increase and is a 40 per cent increase in picture area, which is why the step up looks so much larger in a showroom than the specification suggests.
The cube is steeper still
A sphere of radius 6 has a volume of 904.8 units. Double the radius to 12 and the sphere volume calculator gives 7,238.2, which is exactly eight times.
That is the ratio behind a surprising number of practical facts. A tank twice the size in every dimension holds eight times the water and needs a base that can carry eight times the weight. A scale model at half size has an eighth of the volume and so roughly an eighth of the material cost, which is why model kits are cheap and the full-size object is not.
It also runs backwards. Halving a dimension to save material saves seven eighths of it, not half, which is the kind of saving that makes a design review worthwhile.
Units carry the exponent too
This is where the arithmetic quietly breaks. One metre is 3.2808399 feet, a number many people can recall. One square metre is not 3.28 square feet; it is 10.76391505, because the conversion factor itself gets squared.
The area converter handles it, and the reason to use it rather than a remembered figure is that the error is large and plausible-looking. Convert a 500 square metre floor with the linear factor and you get 1,640 square feet instead of 5,382, which is a mistake you would notice, and converting a small bathroom the same way gives a number that looks wrong but not obviously impossible.
Volume units cube the factor, so a cubic metre is about 35.3 cubic feet rather than 3.28. Area units with their own names hide the relationship completely: a hectare is 2.471 acres, and nothing in either word tells you that both are areas built from a length squared.
Where it shows up in a quote
Paint, tiles, turf, render and plasterboard are all priced by area, so a room 20 per cent larger in both directions needs 44 per cent more material rather than 20. Concrete, soil, gravel and water are priced by volume, so the same 20 per cent in all three dimensions is 73 per cent more.
The habit worth having is to notice which of the three a quantity is before scaling it at all. Ordering material covers the waste allowances that sit on top, and the TV size calculator is the version of this question that most people meet first.