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hscscience Maths Std · Y11
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Module 2 · L12 of 22 50–55 min ⚡ +95 XP available

Scale Drawings and Maps

A scale factor compresses reality into a manageable diagram. Understanding the ratio 1:n, and crucially, how it changes for areas, is the core skill for every map and floor plan problem.

Today's hook, A map has scale 1:50 000. You measure 3.2 cm between two towns. How far apart are they? If a paddock is 2 cm × 3 cm on the map, what is its real area?
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Recall, your gut answer first

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Worksheets

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Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

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Recall, your gut answer first
+5 XP warm-up

A map has a scale of 1:50 000. You measure 3.2 cm between two towns on the map. How far apart are the towns in real life? Also, if a paddock appears as a 2 cm × 3 cm rectangle on the map, what is its real area?

Without calculating make a prediction and explain your reasoning.

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The formulas you need to own
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Scale problems hinge on two key rules: lengths scale by $n$, but areas scale by $n^2$. Forgetting the squared factor is the most common source of errors in HSC scale questions.

A scale of 1:$n$ means every 1 unit on the drawing represents $n$ units in reality. Lengths multiply by $n$; areas multiply by $n^2$. To find the scale, divide the actual measurement by the drawn measurement (in the same units).

Length scales by n; area scales by n².
Actual area $=$ drawn area $\times n^2$, the area scale factor is always $n^2$
Reading a scale 1:n
Actual length = drawn length × $n$. Drawn length = actual ÷ $n$. Units must match on both sides before dividing.
Area scale factor
Lengths scale by $n$, so areas scale by $n^2$. A 1:200 scale means area scales by 40 000. Never forget the square.
Finding the scale
$n = \text{actual} \div \text{drawn}$ (same units). Express as 1:$n$.
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What you'll master
Know

Key facts

  • Scale notation: 1:$n$ means 1 unit drawn = $n$ units actual
  • Linear scale factor $= n$; area scale factor $= n^2$
  • A scale bar is a line on a map representing a real distance
  • Floor plan scales often given in mm:m or cm:m
Understand

Concepts

  • Why area scales as $n^2$ when lengths scale as $n$
  • How to use a scale bar when no numerical scale is given
  • How floor plans and site plans relate to real structures
Can do

Skills

  • Convert between drawn and actual lengths using a scale
  • Convert between drawn and actual areas (using $n^2$)
  • Find the scale ratio given drawn and actual measurements
  • Solve composite floor plan problems
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Key terms
ScaleThe ratio of a drawn measurement to the corresponding real measurement, written 1:$n$ where $n$ is the number of real units per drawn unit.
Scale factorThe number $n$ in a 1:$n$ scale; multiply drawn lengths by $n$ to get actual lengths.
Scale barA line segment drawn on a map or plan, labelled with the real distance it represents; useful when the diagram is reproduced at different sizes.
Floor planAn overhead (bird's-eye) view scale drawing of a building's interior, showing room dimensions and layout.
Area scale factor$n^2$, the factor by which drawn areas are multiplied to get actual areas when the linear scale is 1:$n$.
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Reading a scale, the 1:n notation

A scale of 1:200 means every 1 unit on the drawing represents 200 units in reality.

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Reading a scale, the 1:n notation
core concept

A scale of 1:200 means every 1 unit on the drawing represents 200 units in reality. If you draw 1 cm, the real length is 200 cm = 2 m.

  • To find the actual length: multiply drawn length × $n$
  • To find the drawn length: divide actual length ÷ $n$
  • Units: the scale ratio uses the same units on both sides, convert before dividing
$$\text{Actual length} = \text{drawn length} \times n$$
$$\text{Actual area} = \text{drawn area} \times n^2$$
Area warning. A 1:200 scale means area is scaled by 200² = 40 000. So 1 cm² on the drawing represents 40 000 cm² = 4 m² of actual area. Always square $n$ for areas.

Scale 1:n means 1 unit on the map = n units in reality. Actual length = map length × n. Map length = actual length ÷ n. For area, scale factor is squared: actual area = map area × n². Always state units in the final answer.

Pause, copy the scale rule (1:n means 1 map unit = n real units; actual length = map length × n), the area scale rule (actual area = map area × n²), and the step of always stating units in both map and real dimensions into your book.

Did you get this? True or false: if a scale drawing uses 1:100, then an area on the drawing must be multiplied by 10 000 (not 100) to get the actual area.

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Worked examples · 3 in a row, reveal as you go

Worked examples · 3 in a row, reveal as you go

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · MAP DISTANCES

A map has scale 1:25 000. A road measures 8.4 cm on the map. (a) Find the actual length of the road in km. (b) Another road is 3.5 km long. Find its length on the map in cm.

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(a) Actual $= 8.4 \times 25\,000 = 210\,000 \text{ cm} = 2.1 \text{ km}$
Multiply drawn length by scale factor; convert cm to km (÷100 000).
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PROBLEM 2 · AREA SCALE FACTOR

A floor plan has scale 1:100. A room appears as a 5.2 cm × 3.8 cm rectangle on the plan. (a) Find the actual dimensions of the room. (b) Find the actual area of the room in m².

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(a) Length $= 5.2 \times 100 = 520$ cm $= 5.2$ m; Width $= 3.8 \times 100 = 380$ cm $= 3.8$ m
Multiply each dimension by scale factor 100.
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PROBLEM 3 · FINDING THE SCALE

On a site plan, a fence is drawn as 4.5 cm. The actual fence is 27 m. Find the scale of the drawing.

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Actual $= 27 \text{ m} = 2700 \text{ cm}$
Convert to same units as drawn measurement.
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Quick check: Scale 1:400. Two points are 7.5 cm apart on a drawing. What is the actual distance in metres?

Multiplying a length by n multiplies the corresponding area by n².
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Common errors · the traps that cost marks

Common errors · the traps that cost marks

Common errors · the traps that cost marks

Trap 01
Using n instead of n² for area
Wrong: Multiply drawn area by $n$ to get actual area. Right: Multiply by $n^2$. Lengths scale by $n$; areas always scale by $n^2$. At 1:100, area scales by 10 000, not 100.
Trap 02
Not converting units before finding the scale
When finding $n =$ actual ÷ drawn, both measurements must be in the same unit. Convert actual to centimetres (or drawn to metres) before dividing, or the ratio will be wrong.
Trap 03
Confusing metric area conversions
Wrong: Converting units only requires multiplying by 10. Right: Metric area conversions use powers of 100 (cm² to m²: ÷10 000) and volume uses powers of 1000.

Fill the gap: Scale 1:500. A drawn area of 6 cm² represents an actual area of 6 × ² = 6 × = 1 500 000 cm² = 150 m².

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Quick-fire practice · 9 scale calculations

Quick-fire practice · 9 scale calculations

Quick-fire practice · 9 scale calculations

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A map has scale 1:50 000. A river measures 6.3 cm on the map. Find its actual length in km.

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Scale 1:200. A wall is 8.5 m long. Find its length on the plan in cm.

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Scale 1:400. Two points are 12 cm apart on the drawing. Find the actual distance in metres.

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A drawing shows a road 5 cm long. The actual road is 2 km. Find the scale.

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Scale 1:100. A garden bed appears as 3 cm × 4 cm on a plan. Find its actual area in m².

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Scale 1:500. A field appears as a 4 cm × 6 cm rectangle. Find its actual area in m².

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Scale 1:2500. A lake covers 9 cm² on a map. Find its actual area in m².

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A floor plan uses scale 1:50. A bedroom measures 6 cm × 4.5 cm on the plan. Find the actual dimensions and area of the bedroom.

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A kitchen appears as an L-shape on a 1:80 plan: a 5 cm × 3 cm rectangle joined to a 2 cm × 2 cm rectangle. Find the actual area of the kitchen in m².

Match each scale to its area scale factor:

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Revisit your thinking

Earlier you predicted the real distance and area for the 1:50 000 map.

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Revisit your thinking

Earlier you predicted the real distance and area for the 1:50 000 map. Let's check:

Distance: $3.2 \text{ cm} \times 50\,000 = 160\,000 \text{ cm} = 1.6 \text{ km}$

Paddock area: Drawn area $= 2 \times 3 = 6 \text{ cm}^2$; Actual area $= 6 \times 50\,000^2 = 6 \times 2{,}500{,}000{,}000 \text{ cm}^2 = 15\,000{,}000 \text{ m}^2 = 15 \text{ km}^2$. Or equivalently: actual $= 1 \text{ km} \times 1.5 \text{ km} = 1.5 \text{ km}^2$. (Scale bar approach: 3.2 cm = 1.6 km, so 2 cm = 1 km and 3 cm = 1.5 km; area $= 1 \times 1.5 = 1.5 \text{ km}^2$.)

The area trap is real, always square $n$ for areas.

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Show what you have learned

Multiple choice, then short answer under exam conditions.

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Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.

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Short answer
ApplyBand 43 marks

SA 4. A scale drawing of a house block uses a scale of 1:500. (a) The block is drawn as a 6.4 cm × 4.0 cm rectangle. Find the actual dimensions in metres. (1 mark) (b) Find the actual area of the block in m². (1 mark) (c) Land is sold at $850 per m². Find the value of this block. (1 mark)

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ApplyBand 43 marks

SA 5. A floor plan of an apartment uses scale 1:80. The living area appears as an L-shape formed by a 7 cm × 5 cm rectangle with a 3 cm × 3 cm section removed from one corner. (a) Find the drawn area of the L-shape in cm². (1 mark) (b) Find the actual area of the living space in m². (2 marks)

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AnalyseBand 54 marks

SA 6. A bushwalking map has a scale bar showing that 2 cm on the map = 1 km in reality. (a) Express this scale as a ratio 1:$n$. (1 mark) (b) A trail is measured as 11.4 cm on the map. Find its actual length in km. (1 mark) (c) A national park appears as an irregular shape with area 18 cm² on the map. Find its actual area in km². (2 marks)

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📖 Comprehensive answers (click to reveal)

Drill 1: $6.3 \times 50\,000 = 315\,000$ cm $= 3.15$ km  ·  2: $850 \div 200 = 4.25$ cm  ·  3: $12 \times 400 = 4800$ cm $= 48$ m  ·  4: $2\text{ km} = 200\,000$ cm; $n = 200\,000 \div 5 = 40\,000$; Scale: 1:40 000  ·  5: $3 \times 100 = 3$ m; $4 \times 100 = 4$ m; Area $= 12$ m²  ·  6: $4 \times 500 = 2000$ cm $= 20$ m; $6 \times 500 = 30$ m; Area $= 600$ m²  ·  7: $9 \times 2500^2 = 56\,250\,000$ cm² $= 5625$ m²  ·  8: $6 \times 50 = 3$ m; $4.5 \times 50 = 2.25$ m; Area $= 6.75$ m²  ·  9: Drawn area $= 5 \times 3 + 2 \times 2 = 19$ cm²; Actual $= 19 \times 80^2 = 121\,600$ cm² $= 12.16$ m²

SA 4 (3 marks): (a) $6.4 \times 500 = 3200$ cm $= 32$ m; $4.0 \times 500 = 2000$ cm $= 20$ m → 32 m × 20 m [1]. (b) $32 \times 20 = 640$ m² [1]. (c) $640 \times 850 = \$544\,000$ [1].

SA 5 (3 marks): (a) $7 \times 5 - 3 \times 3 = 35 - 9 = 26$ cm² [1]. (b) $26 \times 80^2 = 26 \times 6400 = 166\,400$ cm² $= 16.64$ m² [2].

SA 6 (4 marks): (a) $1\text{ km} = 100\,000$ cm; $n = 100\,000 \div 2 = 50\,000$; Scale $= 1:50\,000$ [1]. (b) $11.4 \div 2 = 5.7$ km [1]. (c) Area scale factor $= 50\,000^2 = 2.5 \times 10^9$; Actual $= 18 \times 2.5 \times 10^9$ cm² $\div 10^{10} = 4.5$ km² [2].