Map Scales & Scale Factors
Learn how map scales work for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-05). A map scale such as \(1:100\,000\) is a ratio: 1 unit on the map stands for 100 000 of the same unit on the ground, so the second number is the scale factor.
You will see why a map scale is just a ratio, use the scale factor to find real distances and map lengths, write a scale as \(1:x\), read a scale bar, and convert between centimetres, metres and kilometres — what you need to plan a bushwalk, road trip or bike ride from a map.
Theory
A map scale such as \(1:100\,000\) is a ratio that links a length on a map to the real distance on the ground. This Year 12 Standard 1 (NSW) guide shows how to use the scale factor to find real distances and map lengths, write a scale as \(1:x\) and read a scale bar.
A map scale such as \(1:100\,000\) is a ratio: \(1\) unit on the map stands for \(100\,000\) of the same units on the ground. In the scale \(1:x\), the number \(x\) is the scale factor, so \(1\text{ cm}\) on the map is \(x\text{ cm}\) in reality.
To go from map to real, multiply the map length by the scale factor, then convert the centimetres to metres or kilometres. To go from real to map, change the real distance to centimetres and divide by the scale factor. The map length is always the much smaller number.
A scale bar is a ruler printed on the map: each division shows how much real distance one centimetre stands for. Key units: \(1\text{ m}=100\text{ cm}\) and \(1\text{ km}=1000\text{ m}=100\,000\text{ cm}\). This is a core Year 12 Standard 1 (NSW) Ratios and rates skill.
With a map scale of \(1:x\), turn a map length into a real distance by multiplying by the scale factor \(x\):
To go from a real distance to a map length, put the real distance in centimetres and divide:
The unit conversion used most often with maps:
How to solve a map scale problem
- Read the scale \(1:x\) (or the scale bar) and note the scale factor \(x\).
- Decide the direction. Map to real: multiply by \(x\). Real to map: change to centimetres first, then divide by \(x\).
- Convert the units at the end: \(\text{cm}\div 100\) for metres, \(\text{cm}\div 100\,000\) for kilometres.
- Check that the answer makes sense: the real distance should be much bigger than the map length.
Multiply by the scale factor \(50\,000\), then convert centimetres to kilometres.
| \(\text{real}\) | \(=\) | \(50\,000 \times 7\text{ cm}\) |
| \(\) | \(=\) | \(350\,000\text{ cm}\) |
| \(\) | \(=\) | \(350\,000 \div 100\,000\text{ km}\) |
| \(\therefore\ \text{real}\) | \(=\) | \(3.5\text{ km}\) |
The trail is \(3.5\text{ km}\) long.
Change the real distance to centimetres, then divide by the scale factor.
| \(18\text{ km}\) | \(=\) | \(18 \times 100\,000\text{ cm}\) |
| \(\) | \(=\) | \(1\,800\,000\text{ cm}\) |
| \(\text{map}\) | \(=\) | \(1\,800\,000 \div 600\,000\) |
| \(\therefore\ \text{map}\) | \(=\) | \(3\text{ cm}\) |
The towns are \(3\text{ cm}\) apart on the map.
Put both lengths in centimetres, then divide both sides by \(5\).
| \(3\text{ km}\) | \(=\) | \(300\,000\text{ cm}\) |
| \(\text{scale}\) | \(=\) | \(5\text{ cm} : 300\,000\text{ cm}\) |
| \(\) | \(=\) | \((5 \div 5) : (300\,000 \div 5)\) |
| \(\therefore\ \text{scale}\) | \(=\) | \(1 : 60\,000\) |
The scale is \(1:60\,000\).
Each \(1\text{ cm}\) division stands for \(250\text{ m}\).
| \(\text{(i) real}\) | \(=\) | \(6 \times 250\text{ m}\) |
| \(\) | \(=\) | \(1500\text{ m} = 1.5\text{ km}\) |
| \(\text{(ii) }250\text{ m}\) | \(=\) | \(250 \times 100 = 25\,000\text{ cm}\) |
| \(\text{scale}\) | \(=\) | \(1\text{ cm} : 25\,000\text{ cm}\) |
| \(\therefore\ \text{scale}\) | \(=\) | \(1 : 25\,000\) |
(i) \(1.5\text{ km}\); (ii) the scale is \(1:25\,000\).
Common pitfalls
Frequently asked questions
What does a map scale of 1:50 000 mean?
It means 1 unit on the map stands for 50 000 of the same units on the ground. So 1 cm on the map is 50 000 cm in real life, which is 500 m or 0.5 km. The number after the colon is the scale factor.
How do you find the real distance using a map scale?
Measure the distance on the map, multiply it by the scale factor, then change the units. For example 7 cm on a 1:50 000 map is 7 times 50 000, which is 350 000 cm. Divide by 100 000 to get 3.5 km.
How do you work out the map distance from a real distance?
Change the real distance to centimetres, then divide by the scale factor. For example 18 km is 1 800 000 cm, and on a 1:600 000 map that is 1 800 000 divided by 600 000, which is 3 cm.
How many centimetres are in a kilometre?
There are 100 000 centimetres in 1 kilometre. A kilometre is 1000 metres and each metre is 100 centimetres, so multiply 1000 by 100.
How do you write a map scale in the form 1:x?
Write the map length and the real length in the same unit, usually centimetres, then divide both sides by the map length so the left side becomes 1. For example 5 cm for 3 km is 5 cm to 300 000 cm, which simplifies to 1:60 000.
How do you use a scale bar on a map?
Find how much real distance one division of the bar stands for, then count or measure how many divisions fit the map distance and multiply. If each 1 cm division is 250 m, then 6 cm on the map is 6 times 250, which is 1500 m or 1.5 km.