Scale and Scale Factors
A scale such as \(1 : 50\) means every length on a drawing or map is \(\frac{1}{50}\) of the real length. This NSW Year 9 Core maths subtopic applies similar figures to maps, plans and models, and is Stage 5 Core content under NESA outcome MA5‑GEO‑C‑01.
You will explain what a scale means, simplify scales written with units, such as \(1\) cm : \(5\) km becoming \(1 : 500\,000\), and write a scale as \(1 : n\). You will find real distances from a map, like Sydney to Newcastle on a map of NSW, find drawing lengths from real ones, convert answers into the units asked for, work out the scale from a model or a photo of the Burj Khalifa, and find a house's real area from a scale drawing. It is part of Year 9 Core Congruence and Similar Figures.
Every question with a fully worked solution.
- Scale Drawings Watch
Theory
A scale such as \(1 : 150\) links a scale drawing, map or model to the real thing. This Year 9 maths page shows what the two numbers mean, how to simplify a scale written with units, and how to find real lengths, drawing lengths and areas from maps and plans.
A scale drawing, map, plan or model is similar to the real object. Every length is multiplied or divided by the same number.
A scale is written as a ratio, drawing first and real life second, with no units. A scale of \(1 : 150\) means \(1\) unit on the drawing stands for \(150\) of the same unit in real life: \(1\) cm stands for \(150\) cm, and \(1\) mm stands for \(150\) mm.
A scale written with units, such as \(1\) cm : \(4\) m, must be simplified by converting both lengths to the same unit first.
Going from the drawing to real life, multiply by the scale. Going from real life to the drawing, divide. Then convert the answer into the unit the question asks for.
The number \(n\) in \(1 : n\) is the scale factor from the drawing to real life.
For a scale of \(1 : n\), with both lengths in the same unit:
| Convert | Multiply by |
|---|---|
| m to cm | \(100\) |
| cm to mm | \(10\) |
| m to mm | \(1000\) |
| km to m | \(1000\) |
| km to cm | \(100\,000\) |
How to solve a scale problem
- Write the scale as \(1 : n\) with no units, converting first if it has units.
- Decide the direction: drawing to real life, multiply by \(n\); real life to drawing, divide by \(n\).
- Convert the given length into the drawing's unit before dividing, if needed.
- Calculate, then convert the answer into the unit asked for.
- Check that the answer is sensible: a real room is metres long, not kilometres.
Convert so both lengths are in the same unit, then drop the units.
| \(4 \text{ m}\) | \(=\) | \(4 \times 100 \text{ cm} = 400 \text{ cm}\) |
| \(1 \text{ cm} : 4 \text{ m}\) | \(=\) | \(1 \text{ cm} : 400 \text{ cm}\) |
| \(\) | \(=\) | \(\mathbf{1 : 400}\) |
Every length in real life is \(400\) times the length on the drawing.
Drawing to real life: multiply each length by \(150\), then convert to metres. Find the area from the real lengths.
| \(\text{length}\) | \(=\) | \(5 \times 150 = 750 \text{ cm} = 7.5 \text{ m}\) |
| \(\text{width}\) | \(=\) | \(3.2 \times 150 = 480 \text{ cm}\) |
| \(\) | \(=\) | \(4.8 \text{ m}\) |
| \(\text{area}\) | \(=\) | \(7.5 \times 4.8\) |
| \(\) | \(=\) | \(\mathbf{36} \text{ m}^2\) |
Scale each length first. Multiplying the plan area by \(150\) would give the wrong answer.
Real life to map: convert to centimetres first, then divide by \(25\,000\).
| \(7.5 \text{ km}\) | \(=\) | \(7.5 \times 100\,000 \text{ cm}\) |
| \(\) | \(=\) | \(750\,000 \text{ cm}\) |
| \(\text{map length}\) | \(=\) | \(\dfrac{750\,000}{25\,000}\) |
| \(\) | \(=\) | \(\mathbf{30} \text{ cm}\) |
Write model : real, convert to the same unit, then divide both sides so the ratio starts with \(1\).
| \(8 \text{ cm} : 4 \text{ m}\) | \(=\) | \(8 \text{ cm} : 400 \text{ cm}\) |
| \(\) | \(=\) | \(8 : 400\) |
| \(\) | \(=\) | \(\mathbf{1 : 50}\) |
Dividing \(8 : 400\) by \(8\) gives \(1 : 50\). The real car is \(50\) times as long as the model.
Common pitfalls
Frequently asked questions
What does a scale of 1:100 mean?
One unit on the drawing stands for \(100\) of the same unit in real life. So \(1\) cm on the drawing is \(100\) cm, or \(1\) m, in real life.
How do you simplify a scale with units?
Convert both lengths to the same unit, then drop the units and divide so the first number is \(1\). For example, \(1\) cm : \(4\) m is \(1\) cm : \(400\) cm, which is \(1 : 400\).
How do you find the real distance on a map?
Measure the distance on the map and multiply by the scale number. Then convert the answer to metres or kilometres. At \(1 : 50\,000\), \(6\) cm on the map is \(300\,000\) cm, which is \(3\) km.
How do you find the scale of a model?
Write the model length and the real length as a ratio, convert both to the same unit, then divide both by the model length.
How do you find a real area from a scale drawing?
Use the scale to turn each length on the drawing into a real length, then multiply the real lengths together.