Proportion and Scaling
Similar shapes and the scale factor
Two figures are called similar when one is simply a scaled version of the other: they have exactly the same shape, all their corresponding angles are equal, and every length in one is the matching length in the other multiplied by a single number called the scale factor, usually written k. A photograph enlarged on a screen, a model car, a map of a city, each is similar to the real thing. When k is greater than 1 the figure is enlarged; when k is between 0 and 1 it is shrunk. Similarity is one of the most practical ideas in measurement, because it lets us reason about huge or tiny things using manageable scaled copies, provided we know how scaling affects what we are measuring.
Length: scaling as expected
The simplest effect of a scale factor is on length, and it behaves exactly as intuition suggests: every length is multiplied by k. Under a scale factor of 3, a side of 3 units becomes 9, a perimeter triples, a diagonal triples. Because lengths are one-dimensional, they respond directly and proportionally to the scale factor, with no surprises. This is the part of scaling everyone gets right, and it is worth stating clearly precisely so that the contrast with area and volume stands out. Length scales by k, perimeter scales by k, any one-dimensional measurement scales by k. It is the higher dimensions where intuition tends to go wrong.
Area: scaling by the square
Here is the first surprise of scaling. When every length is multiplied by k, area is not multiplied by k but by k². Double all the lengths of a shape and its area becomes four times as large, not twice; triple the lengths and the area grows ninefold. The reason is that area is two-dimensional: an enlarged square fits k rows of k smaller squares, which is k × k in total. This catches people out constantly, whether estimating paint for a wall twice the size, or the cost of a pizza of double the diameter. Remembering that area scales by the square of the scale factor, k², is one of the most useful and most often forgotten facts in measurement.
Volume: scaling by the cube
Volume takes the pattern one step further still, scaling by k³. Double every length and the volume becomes eight times as large; triple them and it grows twenty-sevenfold. Volume is three-dimensional, so an enlarged cube holds k × k × k of the originals, which is k³. The consequences are dramatic and everywhere. A scale model at one-tenth the length has only one-thousandth of the volume, and so one-thousandth of the weight if made of the same material. The same law explains why a giant could not simply be a scaled-up human: weight rises with the cube of height while bone strength rises only with the square, so the giant would be crushed by its own mass. Length, area, volume: k, k², k³.
Scale in the real world
All of this comes together in the everyday use of scale, expressed as a ratio. A map or plan marked 1 : 100 means that one unit on the drawing represents 100 of the same units in reality, so a centimetre on the page is a metre on the ground; a scale of 1 : 25 000 has one centimetre standing for 250 metres. Reading a scale lets you convert distances between the drawing and the real world in either direction. And the dimensional lesson still applies: while distances on a map follow the length scale factor, an area on the ground covered by a region on a 1 : 100 map is larger not by 100 but by 100², that is 10 000 times. Scale ratios make plans, maps and models usable, as long as you respect how length, area and volume each transform.
A scaling problem, modelled from start to finish
Here is the whole modelling cycle on one practical question. A class builds a 1 : 20 scale model of a planned rainwater tank, and the model holds 1.5 litres. To formulate the problem, treat the real tank as exactly similar to the model, so every length is multiplied by k = 20 and the capacity by k cubed, 20 × 20 × 20 = 8000. To solve it, multiply: 1.5 litres × 8000 = 12 000 litres. To interpret the answer, put it back in the situation: 12 000 litres is 12 cubic metres, a believable size for a large rainwater tank, so the result makes sense. To evaluate the model, question its assumption. The real tank’s walls, base and pipes are not scaled copies of the model’s, and thicker walls take up some of the inside, so the true capacity will be a little less. Suppose the real tank measures 2.28 metres high inside while the model measures 12 centimetres. Then the length factor is 228 ÷ 12 = 19, not 20, and the model is modified: 1.5 × 19 × 19 × 19 gives about 10 300 litres. To report, state the scale factor, the similarity assumption, the working and the final estimate with its likely error, so a reader can check every step.