ACARA v9 CONTENT DESCRIPTION “use mathematical modelling to solve practical problems involving proportion and scaling of objects; formulate problems and interpret solutions in terms of the situation; evaluate and modify models as necessary, and report assumptions, methods and findings”
Builds on: Surface Area and Volume of Composite Objects. Working with the area and volume of solids leads to a deeper question: what happens to those quantities when a shape is scaled up or down? This unit examines similarity and scale, and the striking way length, area and volume each respond differently to a scale factor.
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.
Similar shapes: scaled copies
Similar figures have the same shape and equal angles, with every length multiplied by a scale factor.
Two figures are similar if one is a scaled copy of the other: the same shape, with all angles equal and all lengths multiplied by the same number, called the scale factor k. Enlarging or shrinking a rectangle keeps it a rectangle of the same proportions. Similarity is the idea behind every map, model and resized image.
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.
Length scales by k
Under a scale factor k, every length in the figure is multiplied by k.
When a shape is scaled by a factor k, every length is simply multiplied by k. A side of 3 becomes 3 times k. This is the most intuitive part of scaling and the one people expect, but area and volume behave quite differently, which is the key lesson of this unit.
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 k squared. 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 times 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, k squared, is one of the most useful and most often forgotten facts in measurement.
Area scales by k squared
If lengths scale by k, area scales by k squared, so doubling lengths quadruples the area.
Here is the surprise: when lengths scale by k, area scales by k squared. Doubling every length (k = 2) does not double the area, it quadruples it, because the square now fits 2 by 2, that is 4, of the originals. Tripling lengths multiplies area by 9. Area grows with the square of the scale factor, a fact that catches many people out.
Volume: scaling by the cube
Volume takes the pattern one step further still, scaling by k cubed. 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 by k by k of the originals, which is k cubed. 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, k squared, k cubed.
Volume scales by k cubed
If lengths scale by k, volume scales by k cubed, so doubling lengths multiplies volume by eight.
Volume scales even more dramatically: by k cubed. Doubling every length multiplies volume by 8, not 2, because the cube now holds 2 by 2 by 2 of the originals. Tripling lengths multiplies volume by 27. This is why a scale model with half the length has only an eighth of the volume, and why small creatures and large ones are built so differently.
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 to 100 means that one unit on the drawing represents one hundred of the same units in reality, so a centimetre on the page is a metre on the ground; a scale of 1 to 25000 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 to 100 map is larger not by 100 but by 100 squared, that is 10000 times. Scale ratios make plans, maps and models usable, as long as you respect how length, area and volume each transform.
Scale in maps and models
A scale ratio such as 1 to 100 tells you how a length on a drawing relates to the real length.
Scale appears everywhere as a ratio, on maps, plans and models. A scale of 1 to 100 means 1 unit on the drawing stands for 100 of the same units in reality, so 1 centimetre represents 1 metre. Reading a scale lets you convert between drawing and reality in both directions, and the same length scale factor governs how distances on the map relate to true distances.
Quick self-check
1. Two figures are similar when:
2. A shape is enlarged by a scale factor of 3. Each length becomes:
3. If every length of a shape is doubled (k = 2), its area becomes:
4. If every length of a solid is doubled (k = 2), its volume becomes:
5. On a map with scale 1 : 100, a distance of 1 cm on the map represents a real distance of: