Using a Scale

I. What is a scale?

A. Examples and definition

On a map drawn to the scale of \(\frac{1}{10,000,000}\), the distances are 10,000,000 times smaller than the real distances. To draw it, the real distances are multiplied by \(\frac{1}{10,000,000}\), which is the same as dividing them by 10,000,000.

In the same way, on a reproduction of an insect to the scale of 15, the insect is represented 15 times larger than in reality. To draw it, the real dimensions of the insect are multiplied by 15.

B. A little history

The first map of the whole kingdom of France was created at the request of Louis XV, who was impressed by the mapmaking carried out in Flanders. César-François Cassini de Thury, also known as Cassini III, was asked to complete this map on a scale of 1/86,400. The map was based on the grid network created between 1683 and 1744 by his father and grandfather.

The survey began in 1760 and was completed by his son Jacques Dominique Cassini in 1789. The publication of the maps was delayed by the French Revolution and was not completed until 1815. Four generations of Cassinis were devoted to the creation of this map, which was used as a reference by the cartographers of the principal European nations throughout the first half of the 19th century.

II. How is a scale used?

A. If the scale is given by a number

Let's go back to the example of the map created to the scale of \(\frac{1}{10,000,000}\). 

To draw this, the real distances – for example the distance from Paris to Rome as the crow flies –are all multiplied by the same number, here \(\frac{1}{10,000,000}\).

The distance between Paris and Rome as the crow flies is around 1,200 kilometers (km).

The two towns are therefore 0.00012 km apart on the map (since 1,200 (D) × 0.0000001 (e) = 0.00012 (d)). This distance is expressed in kilometers, which is not a very convenient unit to use for a map; converting it into centimeters, it is 12 cm. 

In the case of the reproduction of an insect, the real dimensions (for example, the length of a leg) are all multiplied by the same number: 15. A leg that actually measures 5 millimeters (mm) will measure 75 mm on the drawing/model, that is, 7.5 cm (5 (D) × 15 (e) = 75 (d)).

B. If the scale is given by a comparison of two lengths

For some maps and plans, the scale is given like this: 1 cm to 25 km, or 1 cm to 350 meters (m), etc.

In the first case, 25 km on the ground is represented by 1 cm on the map. This scale factor unlike those we worked with earlier and in the previous lesson consists of two different units. It can be represented as a ratio or a fraction, as shown:

\[e = \frac{1\;\mathrm{cm}}{25\;\mathrm{km}}\]

If something measures 10 cm on the map, we can use the formula \(D \times e = d\) to find the actual distance on the ground:

• Let \(d = 10\;\mathrm{cm}\)

\[D \times \frac{1\;\mathrm{cm}}{25\;\mathrm{km}} = 10\;\mathrm{cm}\]

\[D = \frac{10\;\mathrm{cm}}{\frac{1\;\mathrm{cm}}{25\;\mathrm{km}}} \rightarrow 10\;\mathrm{cm} \div \frac{1\;\mathrm{cm}}{25\;\mathrm{km}}\]

If we apply the technique for the division of fractions, the units will cancel out:

\[\require{cancel} D = 10\;\cancel{\mathrm{cm}} \times \frac{25\;\mathrm{km}}{1\;\cancel{\mathrm{cm}}}\]

\[D = 250\;\mathrm{km}\]

Therefore, if something measures 10 cm on the map, the distance on the ground will be 250 km.