Scale Maps and Plans
A scale such as \(1:50\,000\) means every real length is \(50\,000\) times its length on the map or plan. In NSW Year 8 maths, a scale is treated as a ratio of two lengths written in the same unit (NESA outcome MA4‑RAT‑C‑01).
You will write scales such as \(1\) cm to \(200\) m in the form \(1:n\), read scale bars, and change map distances into real distances and back, moving between millimetres, centimetres, metres and kilometres. You will also find real areas from a floor plan, compare one trail on two maps with different scales, choose the largest scale that fits a page, and find the scale of a half-size photocopy. Bushwalk maps, bedrooms and school ovals all appear. It is part of Year 8 Time and Travel.
Every question with a fully worked solution.
- Scale Maps and Plans - Video - Scale and Distance Watch
Theory
A scale \(1:n\) means every real length is \(n\) times the matching length on the map or plan. On a \(1:20\,000\) map, \(3\) cm stands for \(3 \times 20\,000 = 60\,000\) cm \(= 600\) m. This Year 8 maths page shows how to use scales both ways.
A scale \(1:n\) compares a length on a drawing with the real length: the real length is \(n\) times the drawing length. Both parts are in the same unit, so the scale has no units.
Writing a scale as \(1:n\). Change both lengths to the same unit, then drop the units: \(1\) cm to \(3\) km \(= 1\) cm : \(300\,000\) cm \(= 1:300\,000\). Use \(10\) mm \(= 1\) cm, \(100\) cm \(= 1\) m and \(1000\) m \(= 1\) km.
Drawing to real: multiply by \(n\). Real to drawing: divide by \(n\). Then change to the unit asked for.
A scale bar shows the scale as a picture. Find what \(1\) cm stands for, change it to centimetres, and write \(1:n\).
Finding the scale from one known length: write drawing : real in the same unit and divide both by the drawing length.
Areas. Lengths are \(n\) times bigger, so areas are \(n \times n\) times bigger: on a \(1:500\) plan, \(1\) cm by \(1\) cm stands for \(5\) m by \(5\) m \(= 25\) m\(^2\). Change the lengths first, then find the area.
Comparing scales. The smaller \(n\), the bigger the drawing and the more detail. A half-size copy doubles \(n\). To choose a scale, divide each real length by the space for it and pick the next sensible scale above the larger answer.
Units for scales:
| Change | Multiply by |
|---|---|
| cm to mm | 10 |
| m to cm | 100 |
| km to m | 1000 |
| km to cm | \(100\,000\) |
Using a scale 1:n
- Decide the direction: drawing \(\to\) real (multiply by \(n\)) or real \(\to\) drawing (divide by \(n\)).
- Change units so both lengths are in the same unit (usually cm).
- Calculate, then change to the unit asked for.
| \(400 \times 100\) | \(=\) | \(40\,000\) |
\(400\) m \(= 40\,000\) cm, so the scale is \(1:40\,000\).
(i) What is the real distance for \(3\) cm on the map?
(ii) What is the map distance for a real \(1.2\) km?
| \(3 \times 20\,000\) | \(=\) | \(60\,000 \text{ cm}\) |
| \(1.2 \text{ km}\) | \(=\) | \(120\,000 \text{ cm}\) |
| \(120\,000 \div 20\,000\) | \(=\) | \(6\) |
(i) \(60\,000\) cm \(= 600\) m
(ii) \(6\) cm
(ii) A model uses \(1\) mm for \(5\) m. Write the scale as \(1:n\).
| \(2:500\) | \(=\) | \(1:250\) |
| \(5 \times 1000\) | \(=\) | \(5000\) |
(i) \(5\) m \(= 500\) cm, so \(1:250\)
(ii) \(5\) m \(= 5000\) mm, so \(1:5000\)
(i) Find the real length and width.
(ii) Find the real area.
(iii) Boards cost \(\$65\) per m\(^2\). Find the cost.
| \(8 \times 50\) | \(=\) | \(400 \text{ cm}\) |
| \(5 \times 50\) | \(=\) | \(250 \text{ cm}\) |
| \(4 \times 2.5\) | \(=\) | \(10\) |
| \(10 \times 65\) | \(=\) | \(650\) |
(i) \(4\) m by \(2.5\) m
(ii) \(10\) m\(^2\) (use the real lengths)
(iii) \(\$650\)
(i) Write the scale as \(1:n\).
(ii) How long is a \(2.7\) km trail on the map?
(iii) A bridge is \(4\) mm long on the map. How long is it really?
| \(450 \div 3\) | \(=\) | \(150 \text{ m}\) |
| \(270\,000 \div 15\,000\) | \(=\) | \(18\) |
| \(4 \times 15\,000\) | \(=\) | \(60\,000 \text{ mm}\) |
(i) \(1\) cm stands for \(150\) m \(= 15\,000\) cm: \(1:15\,000\)
(ii) \(18\) cm
(iii) \(60\,000\) mm \(= 60\) m
\(1\) cm stands for \(500\) cm \(= 5\) m, so \(1\) cm\(^2\) stands for \(5 \times 5 = 25\) m\(^2\).
| \(8 \times 25\) | \(=\) | \(200\) |
\(200\) m\(^2\)
Length: \(900 \div 20 = 45\). Width: \(600 \div 15 = 40\). The scale number must be at least \(45\).
| \(900 \div 50\) | \(=\) | \(18\) |
| \(600 \div 50\) | \(=\) | \(12\) |
\(1:50\): the drawing is \(18\) cm by \(12\) cm, which fits. (\(1:40\) gives \(22.5\) cm, too long.)
Common pitfalls
Frequently asked questions
What does a scale of 1:50 000 mean?
Every 1 cm on the map stands for 50 000 cm (500 m) in real life.
How do I find a real distance from a map?
Multiply the map distance by n, then change to metres or kilometres.
How do I find a map distance from a real distance?
Change the real distance to centimetres, then divide by n.
How do I write a scale bar as a ratio?
Find what 1 cm stands for, change it to centimetres, and write 1 : that number.
How do areas change on a scale drawing?
They change by n × n. On a 1:500 plan, 1 square centimetre stands for 25 square metres.