Maps and scale drawings

​​In a nutshell

Many maps and drawings come equipped with a scale, so that you can measure the drawing and use the scale to compute the true distances on the map or dimensions of the drawing. This lesson covers how to read and use scales on maps, or drawings in general.

Reading maps

Most maps are drawn so that north is up. Whenever this is not the case, you should expect the map to feature a compass point indicating the direction of north.

A map must have a scale. You can measure the distance between two points on the map with a ruler, then use the scale to work out what the distance between the corresponding two points in the real world is, via the following procedure:

PROCEDURE

Example 1

On the following map, what is the real distance between the points $$A$$​ and $$B$$?

Use your ruler to measure the length of one of the boxes. The scale given in the bottom right is one box length long.

Measure the distance between $$A$$​ and $$B$$ on your screen with a ruler, and divide it by the length of one of the boxes on your screen (which you just measured), to get that the distance corresponds to $$5$$ box lengths​.

(Alternatively, you could notice that the point $$B$$ is $$3$$ box lengths to the right, and $$4$$ box lengths up, from $$A$$.  Using Pythagoras' theorem, the distance between $$A$$ and $$B$$ on the map is $$\sqrt{3^2 + 4^2} = 5$$ box lengths.)

Using the scale, compute that the real distance between the points $$A$$ and $$B$$ must be:

$$5 \times 10m = 50m$$​​

Therefore, the real distance between the points $$A$$ and $$B$$ is $$\underline{50m}$$.

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Scales as a ratio

Scales can be given as a ratio. Suppose in example $$1$$, you measured one box length to be $$1cm$$. There are $$10 \times 100 = 1000$$​ centimetres in $$10m$$. So, any distance you measured on the map with your ruler would need to be multiplied by $$1000$$ to give you the corresponding distance in the real world. This is a scale of $$1:1000$$​.

In general, the most simplified form of the ratio:

Measured length of scale on map $$:$$ Length written on scale

If the scale given as a ratio. The scale factor is:

$$($$​Length written on scale$$)$$ $$\div$$ $$($$Measured length of scale on map$$)$$​

This is what you multiply the distances you measure on the map by to find the corresponding real world distance.

Scale drawings

Drawings can come equipped with a scale too, and all of the above holds true for drawings the same way it does for maps, including the procedure described for computing real distances between points - just replace the word "map" with drawing.

You might be asked to compute the scale factor of a drawing, to compute the true size of an object from a drawing with a given scale, or you could be asked to produce scale drawings given a scale factor and measurements.

Example 2

The following sketch of a house comes equipped with a scale. How tall is the house in metres? Suppose you measure the length of the scale to be $$2cm$$. What is the scale factor of the drawing?

The house is five blocks tall, and each block length corresponds to a real distance of $$130cm$$, so the real height of the house is:

$$\begin{aligned}130cm \times 5 &= 650cm \\&= 6.5m\end{aligned}$$​​

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If you measure the length of the scale to be $$2cm$$, then the scale factor is given by $$($$length written on scale$$)$$​ $$\div$$​ $$($$​measured length of scale$$)$$​, which is:

​$$\dfrac{130cm}{2cm} = 65$$​​

And as a ratio, the scale is $$1:65$$.

Therefore, the house is $$\underline{6.5m}$$ tall, and the scale factor of the drawing is $$\underline{65}$$.