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.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
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.
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).
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
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
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
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
What to write in your book
- Scale 1:$n$ — actual = drawn × $n$; drawn = actual ÷ $n$.
- Area scale factor = $n^2$. Lengths scale by $n$; areas by $n^2$; volumes by $n^3$.
- Finding the scale: convert to same units, then $n =$ actual ÷ drawn. Express as 1:$n$.
- Always convert units before applying the scale formula.
- Metric conversions: lengths ÷ 10/100/1000; areas ÷ 100/10 000/1 000 000.
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.
Worked examples · 3 in a row, reveal as you go
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.
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.
On a site plan, a fence is drawn as 4.5 cm. The actual fence is 27 m. Find the scale of the drawing.
What to write in your book
- Length: actual = drawn × $n$; drawn = actual ÷ $n$. Always convert to same units first.
- Area: actual area = drawn area × $n^2$. The square is essential — never forget it.
- Scale: $n$ = actual ÷ drawn (same units). Write as 1:$n$.
- L-shaped floor plans: split into rectangles, add (or subtract) areas, then apply $n^2$.
Quick check: Scale 1:400. Two points are 7.5 cm apart on a drawing. What is the actual distance in metres?
Common errors · the traps that cost marks
What to write in your book
- Scale: always convert to same units before finding $n$.
- Area trap: area scale factor is $n^2$, never just $n$.
- 1 m² = 10 000 cm²; 1 km² = 1 000 000 m².
- L-shaped rooms: area of full rectangle minus cut-out, then × $n^2$.
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.
Quick-fire practice · 9 scale calculations
A map has scale 1:50 000. A river measures 6.3 cm on the map. Find its actual length in km.
Scale 1:200. A wall is 8.5 m long. Find its length on the plan in cm.
Scale 1:400. Two points are 12 cm apart on the drawing. Find the actual distance in metres.
A drawing shows a road 5 cm long. The actual road is 2 km. Find the scale.
Scale 1:100. A garden bed appears as 3 cm × 4 cm on a plan. Find its actual area in m².
Scale 1:500. A field appears as a 4 cm × 6 cm rectangle. Find its actual area in m².
Scale 1:2500. A lake covers 9 cm² on a map. Find its actual area in m².
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.
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:
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.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
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 $850 per m². Find the value of this block. (1 mark)
SA 5. A floor plan of an apartment uses scale 1:80. The living area appears as an L-shape formed by a 7 cm × 5 cm rectangle with a 3 cm × 3 cm section removed from one corner. (a) Find the drawn area of the L-shape in cm². (1 mark) (b) Find the actual area of the living space in m². (2 marks)
SA 6. A bushwalking map has a scale bar showing that 2 cm on the map = 1 km in reality. (a) Express this scale as a ratio 1:$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)
📖 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].
Five timed questions on scale drawings and map reading. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaSprint through questions on scale drawings and map reading. Pool: lessons 1–12.
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