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.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
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?
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Come back to this at the end of the lesson.
Wrong: Converting units only requires multiplying by 10.
Right: Metric conversions use powers of 10, but area conversions use powers of 100 and volume uses powers of 1000.
Understanding the Scale Ratio
Always check your units before substituting into formulas. Converting to consistent units is a common source of errors in assessment tasks.
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.
A map has scale 1:25 000. A road measures 8.4 cm on the map.
A floor plan has scale 1:100. A room appears as a 5.2 cm × 3.8 cm rectangle on the plan.
On a site plan, a fence is drawn as 4.5 cm. The actual fence is 27 m. Find the scale of the drawing.
Section A — Length Conversions
Section B — Area Scale Factor
Section C — Floor Plans
$6.3 \times 50\,000 = 315\,000$ cm $= \mathbf{3.15 \text{ km}}$
$850 \div 200 = \mathbf{4.25 \text{ cm}}$
$12 \times 400 = 4800$ cm $= \mathbf{48 \text{ m}}$
$2 \text{ km} = 200\,000$ cm; $n = 200\,000 \div 5 = 40\,000$; Scale: $\mathbf{1:40\,000}$
Actual: $3 \times 100 = 300$ cm $= 3$ m; $4 \times 100 = 4$ m; Area $= 3 \times 4 = \mathbf{12 \text{ m}^2}$
$4 \times 500 = 2000$ cm $= 20$ m; $6 \times 500 = 30$ m; Area $= 20 \times 30 = \mathbf{600 \text{ m}^2}$
$9 \times 2500^2 = 9 \times 6\,250\,000 = 56\,250\,000 \text{ cm}^2 = \mathbf{5625 \text{ m}^2}$
$6 \times 50 = 300$ cm $= 3$ m; $4.5 \times 50 = 225$ cm $= 2.25$ m; Area $= 3 \times 2.25 = \mathbf{6.75 \text{ m}^2}$
Drawn area $= 5 \times 3 + 2 \times 2 = 19 \text{ cm}^2$; Actual $= 19 \times 80^2 = 19 \times 6400 = 121\,600 \text{ cm}^2 = \mathbf{12.16 \text{ m}^2}$
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
Multiple Choice
1 A map has scale 1:20 000. Two towns are 7.5 cm apart on the map. Their actual distance apart is:
? Regarding this topic, 1 A map has scale 1:20 000. Two towns are 7.5 cm apart on the map. Their actual distance apart is:
2 A floor plan uses scale 1:100. A rectangular room appears as 4.2 cm × 3.0 cm on the plan. The actual area of the room is:
? Regarding this topic, 2 A floor plan uses scale 1:100. A rectangular room appears as 4.2 cm × 3.0 cm on the plan. The actual area of the room is:
3 On a drawing, 3 cm represents 12 m. The scale of the drawing is:
? Regarding this topic, 3 On a drawing, 3 cm represents 12 m. The scale of the drawing is:
Short Answer
SA 4 3 marks 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)
$6.4 \times 500 = 3200$ cm $= 32$ m; $4.0 \times 500 = 2000$ cm $= \mathbf{32 \text{ m} \times 20 \text{ m}}$
Area $= 32 \times 20 = \mathbf{640 \text{ m}^2}$
$640 \times 850 = \mathbf{\$544\,000}$
SA 5 3 marks 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)
$7 \times 5 - 3 \times 3 = 35 - 9 = \mathbf{26 \text{ cm}^2}$
Actual $= 26 \times 80^2 = 26 \times 6400 = 166\,400 \text{ cm}^2 = \mathbf{16.64 \text{ m}^2}$
SA 6 4 marks 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)
$1 \text{ km} = 100\,000$ cm; $n = 100\,000 \div 2 = 50\,000$; Scale $= \mathbf{1:50\,000}$
$11.4 \div 2 \times 1 = \mathbf{5.7 \text{ km}}$
Area scale factor $= 50\,000^2 = 2.5 \times 10^9$; Actual $= 18 \times 2.5 \times 10^9 \text{ cm}^2$; Convert: $\div (10^5)^2 = \div 10^{10}$; $= 18 \times 2.5 \times 10^9 \div 10^{10} = 4.5 \text{ km}^2$.
Or: $18 \text{ cm}^2 = 18 \times (0.5 \text{ km})^2 = 18 \times 0.25 = \mathbf{4.5 \text{ km}^2}$
Sprint through questions on scale drawings and map reading. Pool: lessons 1–12.