Mathematics Standard • Year 11 • Module 2 • Lesson 12

Scale Drawings and Maps — Skill Drill

Build fluency with the 1:n scale notation: drawn → actual length, drawn → actual area (×n²), and finding the scale from two known measurements.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 For a scale of 1:n, complete each conversion.

Drawn × ____ = Actual. Actual ÷ ____ = Drawn.

Q1.2 For a scale of 1:n, the area scale factor is ____________ . So 1 cm² of drawing represents ____________ cm² of actual area.

Q1.3 Convert each length to the unit shown.

3 km = ____________ cm. 4.5 m = ____________ cm. 250 000 cm = ____________ km.

Stuck? Revisit lesson § Scale Drawing Formulas — the Length, Area and Scale-ratio panels.

2. Worked example — map distance with unit conversion

Follow each line of working. Every step has a reason on the right.

Problem. A map has scale 1:25 000. A road measures 8.4 cm on the map. Find the actual length of the road in km.

Step 1 — Multiply drawn length by scale factor.

Actual = 8.4 × 25 000 = 210 000 cm

Reason: 1 cm on the map represents 25 000 cm in reality.

Step 2 — Convert cm to km (÷100 000).

210 000 ÷ 100 000 = 2.1 km

Reason: 100 cm in 1 m and 1 000 m in 1 km gives 100 000 cm per km.

Conclusion. The actual road length is 2.1 km.

3. Faded example — fill in the missing steps

A floor plan uses scale 1:100. A room appears as a 5.2 cm × 3.8 cm rectangle on the plan. Find the actual dimensions in metres and the actual area in m². Fill in each blank. 4 marks

Step 1 — Length (multiply by 100):

Length = 5.2 × 100 = ____________ cm = ____________ m

Step 2 — Width: Width = 3.8 × 100 = ____________ cm = ____________ m

Step 3 — Area in m² (multiply the m dimensions):

Area = ____________ × ____________ = ____________ m²

Conclusion. The room is ____________ m × ____________ m with an area of ____________ m².

Stuck? Revisit lesson § Worked Example 2 — Area Scale Factor.

4. Graduated practice — Scale calculations

Show your working in the space below each part. Always state units in the final answer.

Foundation — single-step length conversions (4 questions)

Q	Problem	Answer
4.1 1	Scale 1:200. A wall measures 6 cm on the plan. Find the actual length in cm.
4.2 1	Scale 1:400. A drawing shows 12 cm. Find the actual length in metres.
4.3 1	Scale 1:50 000. A river measures 6.3 cm on the map. Find the actual length in km.
4.4 1	Scale 1:200. A wall is 8.5 m long. Find its length on the plan in cm.

Standard — typical HSC difficulty (6 questions)

Show at least one line of substitution and clearly label your final answer with units.

4.5 A drawing shows a road 5 cm long. The actual road is 2 km. Find the scale of the drawing in the form 1:n. 2 marks

4.6 Scale 1:100. A garden bed appears as 3 cm × 4 cm on a plan. Find its actual area in m². 2 marks

4.7 Scale 1:500. A field appears as a 4 cm × 6 cm rectangle. Find its actual area in m². 2 marks

4.8 Scale 1:50. A bedroom measures 6 cm × 4.5 cm on the floor plan. Find the actual area in m². 2 marks

4.9 On a site plan, a fence is drawn as 4.5 cm. The actual fence is 27 m. Find the scale in the form 1:n. 2 marks

4.10 Scale 1:2 500. A lake covers 9 cm² on a map. Find its actual area in m². 2 marks

Extension — composite shapes and combined steps (2 questions)

4.11 A kitchen on a 1:80 plan is L-shaped: a 5 cm × 3 cm rectangle joined to a 2 cm × 2 cm rectangle. Find the actual area of the kitchen in m². 3 marks

4.12 A bushwalking map has a scale bar showing 2 cm = 1 km. (i) Express this as a ratio 1:n. (ii) A national-park boundary appears as a region of area 18 cm² on the map. Find its actual area in km². 3 marks

Stuck on 4.12? In (ii), 2 cm = 1 km means 1 cm = 0.5 km, so 1 cm² = (0.5 km)² = 0.25 km². Multiply the drawn area in cm² by 0.25 to get km².

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 (1:200) I multiplied the drawn length by 200 (not divided) because the actual is bigger than the drawn.

For 4.2 (1:400) I converted my answer from cm to m by dividing by 100 (not 1000) — m is 100 cm.

For 4.4 (1:200) I converted 8.5 m to cm (×100) before dividing by 200 — both sides must share the same unit.

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Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Drawn vs actual

Drawn × n = Actual. Actual ÷ n = Drawn.

Q1.2 — Area scale factor

Area scale = n². 1 cm² drawn = n² cm² actual.

Q1.3 — Unit conversions

3 km = 300 000 cm. 4.5 m = 450 cm. 250 000 cm = 2.5 km.

Q3 — Faded example (1:100 floor plan room)

Step 1: Length = 5.2 × 100 = 520 cm = 5.2 m.
Step 2: Width = 3.8 × 100 = 380 cm = 3.8 m.
Step 3: Area = 5.2 × 3.8 = 19.76 m².
Conclusion: 5.2 m × 3.8 m, area 19.76 m².

Q4.1 — Scale 1:200, drawn 6 cm

Actual = 6 × 200 = 1 200 cm (or 12 m).

Q4.2 — Scale 1:400, drawn 12 cm

Actual = 12 × 400 = 4 800 cm = 48 m.

Q4.3 — Scale 1:50 000, drawn 6.3 cm

Actual = 6.3 × 50 000 = 315 000 cm = 3.15 km.

Q4.4 — Scale 1:200, actual 8.5 m

8.5 m = 850 cm. Drawn = 850 ÷ 200 = 4.25 cm.

Q4.5 — Find scale (5 cm = 2 km)

2 km = 200 000 cm. n = 200 000 ÷ 5 = 40 000. Scale = 1:40 000.

Q4.6 — Scale 1:100 garden 3 cm × 4 cm

Actual dims = 3 m × 4 m. Area = 3 × 4 = 12 m².

Q4.7 — Scale 1:500 field 4 cm × 6 cm

Actual dims = (4 × 500) cm × (6 × 500) cm = 2 000 cm × 3 000 cm = 20 m × 30 m. Area = 600 m².

Q4.8 — Scale 1:50 bedroom 6 cm × 4.5 cm

Actual = 3 m × 2.25 m. Area = 6.75 m².

Q4.9 — Find scale (4.5 cm = 27 m)

27 m = 2 700 cm. n = 2 700 ÷ 4.5 = 600. Scale = 1:600.

Q4.10 — Scale 1:2 500 lake 9 cm²

Area scale = 2 500² = 6 250 000. Actual = 9 × 6 250 000 = 56 250 000 cm² = 5 625 m².

Q4.11 — L-shaped kitchen, 1:80

Drawn area = 5 × 3 + 2 × 2 = 15 + 4 = 19 cm². Actual = 19 × 80² = 19 × 6 400 = 121 600 cm² = 12.16 m².

Q4.12 — Scale bar 2 cm = 1 km

(i) 1 km = 100 000 cm; 2 cm represents 100 000 cm, so 1 cm represents 50 000 cm — scale = 1:50 000.
(ii) 1 cm = 0.5 km, so 1 cm² = 0.25 km². Actual = 18 × 0.25 = 4.5 km².