Scale Drawings and Maps

Example 1

Below is a scale diagram of a table. We can see the scale is represented by a small measurement interval with id="2908299" role="math" $10cm$ noted next to it. All this means is that in this diagram, every interval of that length represents id="2908298" role="math" $10cm$.

This interval is referred to as the drawing's 'scale.'

Scale diagram of a table example, StudySmarter

In that case, how tall is the table? Well, to do this we just check how many of these intervals make up the height of the table?

Scale diagram of a table 90 cm tall, StudySmarter Originals

In this case, it is nine, therefore the drawing is of a table that is $90cm$ tall.

Example 2

The same concept can be applied to maps. One of the useful things about maps is that they can tell us how far away things are from one another. Take this scale map of four towns. We are given the length of 1 mile on the map with an interval, so we can find out the real-life distance between each town.

Scale map of four towns example, StudySmarter Originals

We can see that $13$ intervals fit between towns A and C, and therefore they are $13$ miles apart.

Scale map of four towns, measuring the distance between towns A and C, StudySmarter Originals

Jemma has a map of her town and wants to see how far away the butcher is from the baker. The scale interval on the map says $0.5$ miles. She measures the scale interval as $1cm$, and measures the distance between them on the map as $4cm$.

From this, Jemma calculates the real-world distance between the butcher and the baker using the formula

$reallifedis\tan ce=\frac{measureddis\tan cefromdiagram}{lengthofscaleinterval}\times scalesize$

$reallifedis\tan ce=\frac{4}{1}\times 0.5$

$reallifedis\tan ce=2miles$

Example 1

The diagram below is drawn to scale a scale of $1:150$, how far is the walk from your house to your friend's house. The measurements are provided for you.

Scale map with ratio scale example, StudySmarter Originals

Solution:

The distance between the two houses can be calculated by finding the total measurement on the diagram, and then multiplying them by the scale factor.

$$\begin{gathered} totalmeasurement=5.2+4 \\ =9.2cm \end{gathered}$$

The scale factor, in this case, is $1500$, the second number in the ratio.

$$\begin{gathered} totaldis\tan ce=9.2cm\times 1500 \\ =13800cm \\ =138m \end{gathered}$$

Example 2

From the scale diagram of the vase below, how tall is the real vase, given that on the diagram it was measured at $10cm$, and given that the scale interval was measured as $1cm$, with a scale value of $3cm$. Furthermore, how could this diagram's scale be expressed as a ratio?

Scale diagram of a vase with scale interval example, StudySmarter Originals

Solution:

To find the real height of the vase we must first find the scale factor. This can be done by considering the scale interval. The scale interval measures as $1cm$ on the diagram, and represents a real-world measurement of $3cm$. Therefore the scale factor is

$\frac{3}{1}=3$

Now, we simply multiply the measured height of the vase in the diagram by the scale factor to obtain the real-world height of the vase.

$10\times 3=30cm$

Finally, expressing the scale as a ratio is a simple case of translating the scale interval into ratio form. For every $1cm$ on the diagram, a measurement in real life will be $3cm.$ Therefore, the ratio will be

$1:3$

Example 3

Below is a diagram drawn to scale of a proposed building. The company that commissioned the building to be made stipulated that it could be no taller than $38m$, and no wider than $24$ metres. Does the proposed design for the building fit these stipulations?

Scale diagram of a proposed building design example, StudySmarter Originals

Solutions:

To check if the building's dimensions fit the company's stipulations, we must first determine the real dimensions of the proposed building. We are given the scale as a ratio, $1:500$. From this, we can determine that the scale factor of the diagram is $500.$

Multiplying the measured height by the scale factor gives us the real-world proposed height of the building.

$$\begin{gathered} height=8cm\times 500 \\ =4000cm \\ =40m \end{gathered}$$

Similarly, we can find the proposed width of the building.

$$\begin{gathered} width=4.2cm\times 500 \\ =2100cm \\ =21m \end{gathered}$$

The proposed height of the building is $40m$, which is not less than $38m$, therefore the height is unacceptable.

The proposed width of the building is $21m$, which is less than $24m$, therefore the width is acceptable.