Introduction

How do architects visualize big ideas? How do people find their way to new places without getting lost? How do sportsmen plan set moves? The answers to all these questions involve scaling.

Here we will be looking at how we use scaling (reduction and enlargement) to represent simplified versions of objects and plans.

Architects make models (of bridges, buildings, and so on), people use maps so as not to get lost, and sportsmen look at set formations on a scaled-down version of their playing field.



These are cases where rough models or sketches are not adequate. An effective model is one which gives as much detail as is required for the situation. So if a building is to have a square base and a height 4.3 times its width, only a properly scaled model will show what this will look like.

When important measurements are kept in proportion like this, we say that the model is a scale version of the original. So if a football pitch's width is half of its length, then a scaled-down version should also have these proportions.

When we make a scale copy of an object, the original and the copy have the same proportions. So, in Fig.4, you see that the dinosaur's tail length is about half the length of the whole skeleton for both the real skeleton and the model. You can see this more clearly by using the animation to zoom in on the model.



To describe how much an object has actually been scaled down (or up), we often use ratios. So the dinosaur model above has a scale of 1:12. This means that for every 1 cm that a particular part of the model measures, the corresponding part of the real skeleton measures 12 cm. This also means that the skeleton is 12 times the length of the model dinosaur, as you can see using the animation.



Let's say you make a scale model of a car and all its measurements are one tenth of the actual measurements.



Below are details of two measurements of the real car.

Whereas models are three-dimensional (3-D) representations of objects, maps are flat (2-D) representations of a plan view of an area. However, we still use

ratio

A ratio compares two quantities. The ratio of a to b is often written a:b. For example, if the ratio of the width to the length of a swimming pool is 1:3, the length is three times the width.ratio to relate maps to real-life dimensions. Maps differ in the amount of detail they contain, but the one feature common to most maps is the fact that they are to scale, whether they are used to direct motorists, or to show weather patterns or the heights of mountain ranges.



The map above has been made at a scale of 1:10,000,000.

It is more usual in mathematics to express

enlargement

An enlargement is a type of transformation in which lengths are multiplied whilst directions and angles are preserved. The transformation is specified by a scale factor of enlargement and a centre of enlargement. For every point in the original shape, the transformation multiplies the distance between the point and the centre of enlargement by the scale factor.enlargement or reduction in terms of scale factors rather than ratios. The

scale factor

The scale factor is the ratio of distances between equivalent points on two geometrically similar shapes.scale factor is the number you multiply the original lengths by to get the lengths on the model. An exact full-size copy of something, therefore, has a scale factor of 1. In the model car example we looked at earlier, the scale factor is 0.1, since each of the measurements of the model car is one tenth of the corresponding measurement in the original.


Click on the figure below to interact with the model.




Fig.7 above shows a car body and a corresponding scale mode. Vary the size of the model using the handle and see how the scale factor changes.



In the model we can see that:





– where the original and model measurements are the lengths of equivalent parts (for instance, the height of the car's body). Rearranging this, we see that we can find the scale factor in the following way:





So, as long as we are given measurements of the original object and the scaled copy, we can work out what the scale factor is. Look at the following example.

A design for an invitation has been scaled down using a photocopier. The original and scaled copy measurements are shown in Fig.8 below.



Follow the steps to work out the scale factor of the enlargement.




Now look at the model in Fig.9 below.


Click on the figure below to interact with the model.

Summary

A scale model or drawing has exactly the same shape as the original object, and all the same proportions, but a different overall size.

The scale of a model can be represented as a ratio, which indicates the relative size of the model and the real object.

Ratios can be used to express the scale of a map. You can use the scale of a map to calculate the real distance between two towns.

Scale factors express the degree of enlargement or reduction.

The scale factor can be calculated using measurements of corresponding lengths on the original and the model, using the equation:




A scale factor of less than 1 reduces the size of an object. A scale factor greater than 1 increases its size.