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MST-12-S2-05 ~35 min ⚡ +90 XP available

Map Scales, Scale Drawings and Building Plans

A scale is a ratio between a drawing and the real world. Read it, use it in both directions, and you can measure a house you have never stood in.

Today's hook, A house plan is drawn at $1:100$. The kitchen bench measures $3.6\text{ cm}$ on the plan and benchtop costs $\$320$ per metre. Can you price the bench without ever visiting the house?

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Orient and prepare

Capture your first estimate and preview scale language.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any question in this focus area.

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Think first, your gut answer

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A house plan is drawn at a scale of $1:100$. The kitchen bench measures $3.6\text{ cm}$ on the plan. Benchtop material costs $\$320$ per metre.

Without calculating exactly write down roughly what the bench will cost, and say what the $100$ in the scale is actually counting.

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What you will be able to do

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Explain why a scale such as $1:100$ is a ratio with no units
Convert a plan length into a real length, and a real length back into a plan length
Use a map scale to find real distances in metres and kilometres
Choose a sensible scale and construct a scale drawing that fits the page
Estimate areas, quantities of material and costs from a building plan

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Preview the vocabulary

Review the essential terms before using them in the worked methods.

Key terms

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Scale: A ratio comparing a length on a drawing with the matching real length, written drawing : real. Like this: a scale of $1:100$ means $1\text{ cm}$ on the plan stands for $100\text{ cm}$ in the building.
Scale factor: The number you multiply a plan length by to get the real length, which is the second term of the scale. Like this: at $1:250$ the scale factor is $250$, so $4\text{ cm}$ on the plan is $4 \times 250 = 1000\text{ cm}$, or $10\text{ m}$.
Scale drawing: A drawing where every length has been reduced (or enlarged) by the same scale factor, so the shape is unchanged. Like this: a $6\text{ m}$ room drawn at $1:50$ appears as a $12\text{ cm}$ line, and a $3\text{ m}$ wall in the same room appears as $6\text{ cm}$.
Building plan: A scale drawing of a floor viewed from above, used to read off room sizes and work out materials. Like this: a bedroom measuring $4\text{ cm}$ by $3\text{ cm}$ on a $1:100$ plan is $4\text{ m}$ by $3\text{ m}$, so it needs $12\text{ m}^2$ of carpet.

Read scales as ratios

Connect map and plan measurements to real dimensions.

A scale is a ratio with the units stripped off

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A scale is written $1:n$ and carries no units at all. That is deliberate. It means one of anything on the drawing stands for $n$ of the same thing in reality. One centimetre stands for $n$ centimetres; one millimetre stands for $n$ millimetres.

So the arithmetic only ever runs in two directions. Going out to the real world you multiply by $n$. Coming back to the drawing you divide by $n$. Convert the units only after that, never before.

$\text{real} = \text{plan} \times n \qquad \text{plan} = \dfrac{\text{real}}{n}$

Worked example 1, plan length to real length

A wall is $4.5\text{ cm}$ on a plan drawn at $1:200$

We are going out to the real world, so multiply.

$4.5 \times 200 = 900\text{ cm}$

Stay in centimetres, the unit we measured in.

$900\text{ cm} = 9\text{ m}$

Convert last. $100\text{ cm}$ in a metre, so divide by $100$.

Real length $= 9\text{ m}$

Check backwards: $9\text{ m} = 900\text{ cm}$ and $900 \div 200 = 4.5\text{ cm}$.

Multiply first, convert second. Converting to metres before applying the scale is the single most common way to lose these marks, because it invites you to divide by $100$ twice.

Worked example 2, real length back to plan length

A room is $12\text{ m}$ long. Draw it at $1:150$

Coming back to the drawing, so divide.

$12\text{ m} = 1200\text{ cm}$

Here the real length is given in metres, so convert it to centimetres before dividing.

$1200 \div 150 = 8$

Divide by the scale factor.

Plan length $= 8\text{ cm}$

Check: $8 \times 150 = 1200\text{ cm} = 12\text{ m}$.

Worked example 3, a map scale

Two towns are $6.4\text{ cm}$ apart on a $1:50\,000$ map

Same rule. Maps just use a much larger scale factor.

$6.4 \times 50\,000 = 320\,000\text{ cm}$

Multiply out to the real world in centimetres.

$320\,000\text{ cm} = 3200\text{ m} = 3.2\text{ km}$

Divide by $100$ for metres, then by $1000$ for kilometres.

Real distance $= 3.2\text{ km}$

Check: $3.2\text{ km} = 320\,000\text{ cm}$ and $320\,000 \div 50\,000 = 6.4\text{ cm}$.

In your book. Write the two arrows: plan $\times n \to$ real, and real $\div n \to$ plan. Underneath, write in one sentence why a scale has no units.

Did you get this? True or false: on a plan drawn at $1:200$, a line $3\text{ cm}$ long represents a real length of $6\text{ m}$.

Construct a scale drawing

Choose a scale, convert consistently and check the result.

Constructing a scale drawing

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When the exam asks you to construct a scale drawing, the hard part is not the drawing. It is choosing a scale that makes the whole thing fit the space you have been given.

Work out how many times the real object must shrink to fit, in both directions, then round the scale factor up to a friendly number. Rounding up shrinks the drawing further, so it still fits. Rounding down does not.

Worked example 4, choosing a scale

Fit a room $6\text{ m}$ by $4\text{ m}$ into a box $15\text{ cm}$ by $10\text{ cm}$

Both directions have to fit, so test both.

$600 \div 15 = 40$ and $400 \div 10 = 40$

Convert the real lengths to centimetres, then divide by the space available.

$1:40$ fills the box exactly

Take the larger of the two answers. Here they happen to be equal.

Choose $1:50$, giving $12\text{ cm}$ by $8\text{ cm}$

A friendly scale with a margin. $600 \div 50 = 12$ and $400 \div 50 = 8$, both inside the box.

Every length uses the same factor. A drawing where the walls are scaled but a door or a window is not is no longer a scale drawing, and it will not earn the mark.

Digital tools do the arithmetic for you but not the thinking. In a drawing program you still set the scale first, then enter real measurements. On paper you divide each real length by the scale factor, then rule it. The two methods must give the same drawing, and that is the check.

In your book. Sketch the $6\text{ m}$ by $4\text{ m}$ room at $1:50$ using a ruler, then label each side with both the plan length and the real length.

Quick check. A wall is $7.5\text{ m}$ long. At a scale of $1:250$, how long is it on the plan?

Calculate from plans

Use dimensions from a plan to find quantities and costs.

Quantities and costs from a plan

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This is where the marks are. You measure a room on the plan, scale it up, find an area, then multiply by a rate. Every step is one you already know; the question just stacks them.

Worked example 5, floor area from a plan

A room measures $5.2\text{ cm}$ by $3.5\text{ cm}$ at $1:100$

Scale each side separately, before doing anything with area.

$5.2 \times 100 = 520\text{ cm} = 5.2\text{ m}$

At $1:100$ the centimetre reading becomes the metre reading, which is why the scale is so common.

$3.5 \times 100 = 350\text{ cm} = 3.5\text{ m}$

The other side, same factor.

Area $= 5.2 \times 3.5 = 18.2\text{ m}^2$

Multiply the two real lengths. Never the plan lengths.

Worked example 6, costing the floor

Tiles cost $\$45$ per square metre

A rate, so multiply it by the number of square metres.

$18.2 \times 45 = 819$

Use the real area from worked example 5.

Cost $= \$819$

Check: $819 \div 45 = 18.2$, the area we started with.

Scale the sides, not the area

Doubling every length multiplies the area by four. Convert each length first, then multiply.

Read the rate's unit

Per metre needs a length; per square metre needs an area. The unit tells you which one to find.

Estimate is not a free pass

An estimate still has to be worked, then rounded. Say what you rounded and why.

In your book. Copy the four-line chain: measure the plan, multiply by the scale factor, find the area, multiply by the rate. Put the hook question beside it and finish it.

Fill the gaps. A bedroom measures $4.4\text{ cm}$ by $3.0\text{ cm}$ on a plan drawn at $1:50$. Real dimensions m by m, floor area m$^2$, and carpet at $\$40$ per square metre costs $\$$.

Show what you can do

Answer the exam-style questions, then compare your working with the model answers.

Quick-check drill

work before revealing answers

Choose an option for each fixed drill question, then reveal the concept-labelled explanations below.

Drill 1. A plan is drawn at $1:250$. A line measures $6\text{ cm}$ on the plan. What is the real length?

$15\text{ m}$
$1.5\text{ m}$
$150\text{ m}$
$41.7\text{ m}$

Drill 2. A fence is $20\text{ m}$ long. How long is it on a plan drawn at $1:400$?

$5\text{ cm}$
$8\text{ cm}$
$50\text{ cm}$
$0.5\text{ cm}$

Drill 3. On a map with a scale of $1:25\,000$, two points are $8\text{ cm}$ apart. What is the real distance?

$2\text{ km}$
$20\text{ km}$
$0.2\text{ km}$
$200\text{ m}$

Drill 4. Four plans of the same building use different scales. For the same length measured on the plan, which scale gives the largest real length?

$1:500$
$1:200$
$1:100$
$1:50$

Drill 5. A room measures $4.0\text{ cm}$ by $2.5\text{ cm}$ on a plan drawn at $1:100$. What is its floor area?

$10\text{ m}^2$
$1000\text{ m}^2$
$0.1\text{ m}^2$
$100\text{ m}^2$

Short answer

ApplyBand 43 marks

SA 1. A floor plan is drawn at a scale of $1:150$. (a) A corridor measures $9.4\text{ cm}$ on the plan. Find its real length in metres. (2 marks) (b) A doorway is $0.9\text{ m}$ wide in the building. How wide is it on the plan, in centimetres? (1 mark)

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ApplyBand 43 marks

SA 2. Two towns are $7.5\text{ cm}$ apart on a map with a scale of $1:100\,000$. (a) Find the real distance between them, in kilometres. (2 marks) (b) A cyclist averages $18\text{ km/h}$. How long would the ride take, in minutes? (1 mark)

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AnalyseBand 54 marks

SA 3. On a building plan drawn at $1:50$, a rectangular room measures $8.4\text{ cm}$ by $6.0\text{ cm}$. (a) Find the real dimensions of the room, in metres. (2 marks) (b) Find the floor area, in square metres. (1 mark) (c) Carpet costs $\$38$ per square metre. Find the cost of carpeting the room. (1 mark)

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📖 Comprehensive answers (click to reveal)

Plan length to real length. $6 \times 250 = 1500\text{ cm}$, and $1500 \div 100 = 15\text{ m}$. Option B divides by $100$ one time too many. Option D divides by the scale factor instead of multiplying.

Real length to plan length. $20\text{ m} = 2000\text{ cm}$, and $2000 \div 400 = 5\text{ cm}$. Option C forgets to convert the metres to centimetres before dividing.

Map distance conversion. $8 \times 25\,000 = 200\,000\text{ cm} = 2000\text{ m} = 2\text{ km}$. Option D stops at metres and misreads the unit.

Comparing scale detail. Real length $=$ plan length $\times n$, so the largest $n$ gives the largest real length, and $500$ is the largest here. A common error is to read $1:50$ as the largest because $50$ looks like the most detailed drawing, which it is, but detail and real length run opposite ways.

Floor area from a plan. $4.0 \times 100 = 400\text{ cm} = 4\text{ m}$ and $2.5 \times 100 = 250\text{ cm} = 2.5\text{ m}$, so the area is $4 \times 2.5 = 10\text{ m}^2$. Option B scales the area instead of each side.

SA 1 (3 marks): (a) $9.4 \times 150 = 1410\text{ cm}$ [1], which is $14.1\text{ m}$ [1]. (b) $0.9\text{ m} = 90\text{ cm}$, and $90 \div 150 = 0.6\text{ cm}$ [1].

SA 2 (3 marks): (a) $7.5 \times 100\,000 = 750\,000\text{ cm}$ [1], which is $7500\text{ m}$, or $7.5\text{ km}$ [1]. (b) Time $= 7.5 \div 18 = 0.41\dot{6}\text{ h}$, and $0.41\dot{6} \times 60 = 25$ minutes [1]. Check: $18 \times \tfrac{25}{60} = 7.5\text{ km}$.

SA 3 (4 marks): (a) $8.4 \times 50 = 420\text{ cm} = 4.2\text{ m}$ and $6.0 \times 50 = 300\text{ cm} = 3.0\text{ m}$ [2]. (b) Area $= 4.2 \times 3.0 = 12.6\text{ m}^2$ [1]. (c) Cost $= 12.6 \times 38 = \$478.80$ [1]. Check: $478.80 \div 38 = 12.6$.

Retrieve, reflect and finish

Revisit your opening idea, then use the topic challenge and mark the lesson complete.

Revisit your thinking

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Go back to what you wrote in section 01. The bench is $3.6\text{ cm}$ at $1:100$, so it is $360\text{ cm}$, or $3.6\text{ m}$, and at $\$320$ per metre it costs $3.6 \times 320 = \$1152$.

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Focus area overview · Year 12 · Maths Standard