Year 9 Maths Advanced Congruence and Similar Figures

Scale Diagrams

Q: What does a scale of 1 : 100 mean?

It means 1 unit on the drawing stands for 100 of the same units in real life. For example, 1 cm on the drawing is 100 cm, or 1 m, in real life.

Q: How do you find the actual distance on a map?

Measure the map distance, multiply it by n from the scale 1 : n, then convert the answer to metres or kilometres.

Q: What is a scale of 20 : 1?

It is an enlargement. The drawing is 20 times as large as the real object, so divide drawing lengths by 20 to get real lengths.

20 practice questions 1 video lesson Theory + worked examples

A scale such as $1 : 25\,000$ means one unit on the drawing stands for $25\,000$ of the same unit in real life. This NSW Year 9 Advanced maths subtopic uses scales on maps, plans and models (NESA outcome MA5‑GEO‑C‑01).

You will write scales in the form $1 : n$, convert map distances to real distances and back, and work with enlargements such as a SIM card drawn at $20 : 1$. You will find a scale from a photo, the real area of a house, the new scale after a photocopier enlargement, a scale from two areas, the time for a bushwalk, and an unknown scale from two maps. It is part of Year 9 Advanced Congruence and Similar Figures.

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Scale Diagrams - Video - Scale Drawings Watch

Theory

A scale diagram is a similar copy of a real object: every length is multiplied or divided by the same number. A scale of $1 : n$ means $1$ unit on the drawing stands for $n$ of the same units in real life. This Year 9 Advanced maths page covers maps, plans, models and areas.

A scale drawing, map or model is similar to the real object. The scale is the ratio drawing length : real length, with both in the same units.

A scale of $1 : n$ is a reduction: the real object is $n$ times as big. A scale of $n : 1$ is an enlargement, used for tiny things: the drawing is $n$ times as big as the real object.

A scale may be given with units, like $1$ cm : $5$ m. Change it to one unit to write it as $1 : n$: $1$ cm : $500$ cm, so $1 : 500$. A scale bar shows the same information as a picture.

Unit facts: $1$ cm $= 10$ mm, $1$ m $= 100$ cm, $1$ km $= 1000$ m $= 100\,000$ cm. For areas, $1$ m$^2 = 10\,000$ cm$^2$ and $1$ ha $= 10\,000$ m$^2$.

With scale $1 : n$, one unit on the drawing stands for $n$ of the same units in real life.

A scale bar shows the scale as a picture. Here $2$ cm $= 500$ m $= 50\,000$ cm, so the scale is $1 : 25\,000$.

With a scale of $1 : n$:

\[\text{real length} = n \times \text{drawing length} \qquad \text{drawing length} = \dfrac{\text{real length}}{n}\]

real = n \times drawing

Finding the scale from a matching pair (same units):

\[n = \dfrac{\text{real length}}{\text{drawing length}}\]

n = \frac{real length}{drawing length}

Areas scale by the square:

\[\text{real area} = n^2 \times \text{drawing area}\]

real area = n^{2} \times drawing area

Changing the size of a plan. If a plan is copied at a percentage, each drawing length changes but still stands for the same real length. Work out the new drawing length for \(n\) real units, then rewrite as \(1 : \text{new } n\).

How to use a scale

Same units first: convert the scale and the lengths so every length is in one unit (usually cm or mm).
Drawing to real: multiply by $n$. Real to drawing: divide by $n$.
Convert the answer into the unit asked for (km for maps, m for buildings).
Check it is sensible: a real room is a few metres, a real car about $4$ m.

Writing and using scales

Find a scale: write drawing : real in the same unit, then divide both sides by the drawing length. Round $n$ if asked (e.g. to two significant figures).
Ratio of two sizes: same units, then simplify, e.g. $400$ cm : $5$ cm $= 80 : 1$.
Areas: scale each length, then multiply; or multiply the area by $n^2$. Given two areas, the scale is $1 : \sqrt{\text{area ratio}}$.
Further steps: use the real distance for speed and time ($t = d \div s$), or set up an equation when the scale contains an unknown.

Example 1 — Finding a scale

On a house plan, a wall that is $6.3$ m long is drawn $4.2$ cm long.
(i) Write the scale in the form $1 : n$.
(ii) A doorway is $6$ mm wide on the plan. How wide is the real doorway, in metres?

Solution

(i) Write both lengths in centimetres, then divide both sides by the drawing length.

$6.3\text{ m}$	$=$	$630\text{ cm}$
$4.2 : 630$	$=$	$1 : (630 \div 4.2)$
	$=$	$\mathbf{1 : 150}$

(ii) Plan to real: multiply by $150$. Millimetres stay millimetres.

$6 \times 150$	$=$	$900\text{ mm}$
	$=$	$\mathbf{0.9}\text{ m}$

4.2 : 630 = 1 : 150

Example 2 — Reading a scale bar

A map has the scale bar shown, where $4$ cm stands for $2$ km.
(i) Write the scale as $1 : n$.
(ii) A lake is $11.3$ cm long on the map. How long is it in kilometres?
(iii) Two lookouts are $8.2$ km apart. How far apart are they on the map?

Solution

(i) Change $2$ km to centimetres: $2 \times 1000 \times 100 = 200\,000$ cm.

$4 : 200\,000$

$=$

$\mathbf{1 : 50\,000}$

(ii) Map to real: multiply, then convert ($1$ km $= 100\,000$ cm).

$11.3 \times 50\,000$	$=$	$565\,000\text{ cm}$
	$=$	$\mathbf{5.65}\text{ km}$

(iii) Real to map: convert to centimetres, then divide.

$8.2\text{ km}$	$=$	$820\,000\text{ cm}$
$820\,000 \div 50\,000$	$=$	$\mathbf{16.4}\text{ cm}$

4 : 200000 = 1 : 50000

Example 3 — Area and cost from a plan

A floor plan has a scale of $1 : 50$. A bedroom is drawn $7.6$ cm by $6.4$ cm. Carpet costs $52.50 per square metre. Find the cost to carpet the room.

Solution

Scale each length first, then find the area of the real room.

$7.6 \times 50$	$=$	$380\text{ cm} = 3.8\text{ m}$
$6.4 \times 50$	$=$	$320\text{ cm} = 3.2\text{ m}$
$A$	$=$	$3.8 \times 3.2 = 12.16\text{ m}^2$
$\text{cost}$	$=$	$12.16 \times 52.50 = \mathbf{\$638.40}$

A = 3.8 \times 3.2 = 12.16

Example 4 — Areas on a scale model

A garden is modelled at a scale of $1 : 40$. The real lawn has an area of $72$ m$^2$. Find the area of the model lawn in cm$^2$.

Solution

Every length is divided by $40$, so every area is divided by $40 \times 40 = 40^2 = 1600$.

$72\text{ m}^2$	$=$	$72 \times 10\,000 = 720\,000\text{ cm}^2$
$\text{model area}$	$=$	$720\,000 \div 1600$
	$=$	$\mathbf{450}\text{ cm}^2$

Check with lengths: a $12$ m by $6$ m lawn is $30$ cm by $15$ cm on the model, and $30 \times 15 = 450$ $\checkmark$

720000 \div 40^{2} = 450

Common pitfalls

Convert before you divide. In $2$ cm : $5$ m, change $5$ m to $500$ cm first. A scale $1 : n$ has no units because both sides are in the same unit.

Multiply or divide? Map to real gets bigger, so multiply. Real to map gets smaller, so divide. For an enlargement $n : 1$, it is the other way round.

Areas do not scale by $n$. A scale of $1 : 50$ makes areas $50^2 = 2500$ times as big. Scale the lengths first if in doubt.

Watch the zeros. $1$ km is $100\,000$ cm. Write numbers with thin spaces in groups of three so you can count the zeros.

Frequently asked questions

What does a scale of 1 : 100 mean?

It means $1$ unit on the drawing stands for $100$ of the same units in real life. For example, $1$ cm on the drawing is $100$ cm, or $1$ m, in real life.

How do you find the actual distance on a map?

Measure the map distance, multiply it by $n$ from the scale $1 : n$, then convert the answer to metres or kilometres.

How do you write a scale in the form 1 : n?

Write both lengths in the same unit, then divide both sides by the drawing length so the first number is $1$.

What is a scale of 20 : 1?

It is an enlargement. The drawing is $20$ times as large as the real object, so divide drawing lengths by $20$ to get real lengths.

How do you find a real area from a scale drawing?

Scale each length to its real size first, then work out the area. Or multiply the drawing area by $n^2$, since areas scale by the square of $n$.

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\(6.3\text{ m}\)	\(=\)	\(630\text{ cm}\)
\(4.2 : 630\)	\(=\)	\(1 : (630 \div 4.2)\)
\(\)	\(=\)	\(\mathbf{1 : 150}\)

\(6 \times 150\)	\(=\)	\(900\text{ mm}\)
\(\)	\(=\)	\(\mathbf{0.9}\text{ m}\)

\(11.3 \times 50\,000\)	\(=\)	\(565\,000\text{ cm}\)
\(\)	\(=\)	\(\mathbf{5.65}\text{ km}\)

\(8.2\text{ km}\)	\(=\)	\(820\,000\text{ cm}\)
\(820\,000 \div 50\,000\)	\(=\)	\(\mathbf{16.4}\text{ cm}\)

\(7.6 \times 50\)	\(=\)	\(380\text{ cm} = 3.8\text{ m}\)
\(6.4 \times 50\)	\(=\)	\(320\text{ cm} = 3.2\text{ m}\)
\(A\)	\(=\)	\(3.8 \times 3.2 = 12.16\text{ m}^2\)
\(\text{cost}\)	\(=\)	\(12.16 \times 52.50 = \mathbf{\$638.40}\)

\(72\text{ m}^2\)	\(=\)	\(72 \times 10\,000 = 720\,000\text{ cm}^2\)
\(\text{model area}\)	\(=\)	\(720\,000 \div 1600\)
\(\)	\(=\)	\(\mathbf{450}\text{ cm}^2\)

Scale Diagrams

📖 Theory

How to use a scale

Writing and using scales

Common pitfalls

Frequently asked questions

Theory