Scale Drawings and Maps

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Scale diagrams and drawings are something that everyone will have come across in their lives at some point. If you have ever looked at a map, then you have looked at a scale drawing! They are incredibly useful for representing things on a page that in real life are far bigger or smaller.

In this article, we will discuss what a scale drawing is, give you some examples of this, and show you a formula you can use and its relation to ratios.

What is a scale drawing?

A scale drawing is simply an image representing something in real life that is much larger or smaller, whilst keeping the proportions intact.

What we have to remember about scale diagrams, is that the relative proportions of the diagram are the same as the real-life object.

Take, for instance, an example of a box. If the real-life height of the box is twice the real-life length of the box, then the height of the box in the scale diagram will also be twice the width of the box in the scale diagram.

In other words, a scale drawing keeps the exact same shape as the original subject, just smaller or bigger!

Still not sure? Let's take a look at some examples.

Scale Drawings and Maps Examples

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" $10 c m$ noted next to it. All this means is that in this diagram, every interval of that length represents id="2908298" role="math" $10 c m$ .

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

scale diagrams and maps example of a scale diagram studysmarter 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 diagrams and maps scale diagram example study smarter 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 $90 c m$ 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 diagrams and maps scale map example studysmarter 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 diagrams and maps scale map example studysmarter Scale map of four towns, measuring the distance between towns A and C, StudySmarter Originals

The examples above have hopefully made clearer what we mean by scale diagrams and maps, but usually there won't be a neat little diagram with the intervals lined up between two points for us to count.

So how exactly do we work out real-life measurements from these diagrams? Let's take a look at how we can do that practically with nothing but a ruler and a handy formula!

Scale Drawings and Maps Formula

If we have a scale diagram or map, and we want to discern a certain real-life measurement from it, all we have to do is take the desired measurement from the diagram, and relate it to the real world via the given scale. We can do this in a few easy steps.

Step 1: Use a ruler to measure the scale interval on the diagram.

Step 2: Use a ruler to take the measurement on the diagram that you would like to know.

Step 3: Apply the formula below.

$r e a l l i f e d i s \tan c e = \frac{m e a s u r e m e n t f r o m d i a g r a m}{l e n g t h o f s c a l e i n t e r v a l} \times s c a l e s i z e$

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 $1 c m$ , and measures the distance between them on the map as $4 c m$ .

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

$r e a l l i f e d i s \tan c e = \frac{m e a s u r e d d i s \tan c e f r o m d i a g r a m}{l e n g t h o f s c a l e i n t e r v a l} \times s c a l e s i z e$

$r e a l l i f e d i s \tan c e = \frac{4}{1} \times 0.5$

$r e a l l i f e d i s \tan c e = 2 m i l e s$

This form of the formula is intuitive to how we work out the real-life measurements from scale diagrams, but we can simplify it further to introduce an important aspect of scale diagrams, the scale factor.

Scale Factors of Scale Diagrams and Maps

Starting off with our original formula

$r e a l l i f e d i s \tan c e = \frac{m e a s u r e d d i s \tan c e f r o m d i a g r a m}{l e n g t h o f s c a l e i n t e r v a l} \times s c a l e s i z e$

We can rearrange it to the following form

r e a l l i f e d i s \tan c e = \frac{s c a l e s i z e}{l e n g t h o f s c a l e i n t e r v a l} \times m e a s u r e d d i s \tan c e f r o m d i a g r a m

This form gives the relationship between measurements on the diagram, and measurements in real life, in terms of the scale factor.

$r e a l l i f e d i s \tan c e = s c a l e f a c t o r \times m e a s u r e d d i s \tan c e f r o m d i a g r a m$

The scale factor is just the ratio between the size of something in real life, to the size of that thing on the diagram. As such, the scale factor can be obtained by simply dividing the scale size by the length of the scale interval.

$s c a l e f a c t o r = \frac{s c a l e s i z e}{l e n g t h o f s c a l e i n t e r v a l}$

The scale factor of a scale diagram is the ratio between the actual measurements of something, and the measurements on the scale diagram.

Any real-life measurement can be obtained by multiplying the measurement on the diagram, and then multiplying by the scale factor.

It's possible at this point that you are wondering why on Earth would the people making scale diagrams not just include this ratio on the diagram? Well, the good news is they very often do! Working with them is simple if you know how; it's a good thing we're here.

Scale Drawings Ratio

Ratio scales are often useful on physical scale drawings, where the size of the image does not depend on things such as the size of the device you are viewing it on.

The example of the car below has a scale of $1 : 90$ , that means that for every $1$ centimetres on the diagram there are $90$ centimetres in real life. Equally, it means for every $1$ millimetre on the diagram there are $90$ millimetres, or really any other unit of length. The ratio is not concerned with what unit you use, only the relative size of the diagram compared to the real-life car.

Scale diagrams and maps scale diagrams with ratios car studysmarter Scale diagram of a car with a ratio scale, StudySmarter Originals

So, if we wanted to know the true length of the car, we would simply measure the length on the diagram.

Scale diagrams and maps scale diagrams with ratios car studysmarter Scale diagram of a car showing it as 5cm long, StudySmarter Originals

Since the car in the diagram is $5 c m$ , the car in real life must be $90 \times 5 c m$ , i.e. $4.5 m .$ Do you notice anything about the ratio and the numbers used in that calculation?

Well, it's the same calculation we did earlier with the scale fact. In fact, the second number in the ratio is the scale factor!

Let's put everything we've learned into practice with some examples.

Scale Drawings and Maps Calculation Examples

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 diagrams and maps, scale diagrams with ratio scale example studysmarter 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.

$t o t a l m e a s u r e m e n t = 5.2 + 4 = 9.2 c m$

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

$t o t a l d i s \tan c e = 9.2 c m \times 1500 = 13800 c m = 138 m$

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 $10 c m$ , and given that the scale interval was measured as $1 c m$ , with a scale value of $3 c m$ . Furthermore, how could this diagram's scale be expressed as a ratio?

Scale diagrams and maps height of a vase example studysmarter 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 $1 c m$ on the diagram, and represents a real-world measurement of $3 c m$ . 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.

$10 \times 3 = 30 c m$

Finally, expressing the scale as a ratio is a simple case of translating the scale interval into ratio form. For every $1 c m$ on the diagram, a measurement in real life will be $3 c m .$ 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 $38 m$ , and no wider than $24$ metres. Does the proposed design for the building fit these stipulations?

Scale diagrams and maps scale diagram of a building example studysmarter 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.

$h e i g h t = 8 c m \times 500 = 4000 c m = 40 m$

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

$w i d t h = 4.2 c m \times 500 = 2100 c m = 21 m$

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

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

Scale diagrams and maps - Key takeaways

Scale diagrams are diagrams drawn to be proportionally smaller or larger than their real-life subject.
Scale diagrams will either have a scale interval or ratio scale, with which the size of the actual subject can be calculated using measurements from the diagram.
The scale factor of a scale diagram is a number which relates any measurement from the real world object and the same measurement in the diagram.
When a measurement from the diagram is multiplied by the scale factor, the result is the same measurement in the real-life subject.

Frequently Asked Questions about Scale Drawings and Maps

The scale is a piece of information included in scale drawings or maps that relates the size of the drawing to the size of the real-life subject of the drawing.

Scale drawings and maps are used to represent real-world subjects in a way that keeps their proportionality. This means that these drawings and maps can be used to directly find the actual size of the subject in the real world. This could be the dimensions of an object or place, or to find distances between locations and much more.

The easiest way to perform scale drawing calculations is to find the scale factor, then take a measurement of the thing you would like to know the real-world measurement of. The real-world measurement is found by multiplying these two values.

Almost any map that you pick up will be drawn to scale, as well as things such as blueprints for buildings, or designs for products, and much more.

Scale maps are a type of scale drawing. A scale drawing is any drawing representing something else, that retains the relative proportions of the original subject. In other words, a drawing of something that is not the same size, but is exactly the same shape.

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Scale Drawings and Maps

What is a scale drawing?

Scale Drawings and Maps Examples

Scale Drawings and Maps Formula

Scale Factors of Scale Diagrams and Maps

Scale Drawings Ratio

Scale Drawings and Maps Calculation Examples

Scale diagrams and maps - Key takeaways

Frequently Asked Questions about Scale Drawings and Maps

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Scale Drawings and Maps

What is a scale drawing?

Scale Drawings and Maps Examples

Scale Drawings and Maps Formula

Scale Factors of Scale Diagrams and Maps

Scale Drawings Ratio

Scale Drawings and Maps Calculation Examples

Scale diagrams and maps - Key takeaways

Frequently Asked Questions about Scale Drawings and Maps

What is a scale in a scale drawing or map?

What is the purpose of scale drawings and maps?

How do you calculate scale drawings?

What is an example of scale drawings and maps?

What is the relationship between scale drawings and maps?

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