R2: Scale Drawings
Use scale factors, scale diagrams and maps
📋 Key Concepts
Definition: A scale drawing is a reduced or enlarged drawing that represents an actual object or distance. The scale tells you the ratio between the drawing and real life.
Common Scales
| Scale | Meaning | Use |
|---|---|---|
| 1 : 100 | 1 cm represents 100 cm (1 m) | House plans |
| 1 : 1000 | 1 cm represents 1000 cm (10 m) | Site plans |
| 1 : 25000 | 1 cm represents 250 m | OS Maps |
| 1 : 50000 | 1 cm represents 500 m | OS Maps |
Real Distance = Map Distance × Scale Factor
Map Distance = Real Distance ÷ Scale Factor
Map Distance = Real Distance ÷ Scale Factor
📝 Reading Scales
Understanding Scale: A scale of 1 : 50 means every 1 cm on the drawing represents 50 cm in real life.
Example 1
A map has a scale of 1 : 25000. A distance on the map measures 4 cm. What is the real distance?
Solution:
Real distance = 4 × 25000 = 100000 cm
Convert to km: 100000 ÷ 100000 = 1 km
Or: 4 × 250 m = 1000 m = 1 km
Example 2
A scale drawing uses a scale of 1 : 20. A wall in real life is 3.5 m long. How long is it on the drawing?
Solution:
Convert: 3.5 m = 350 cm
Drawing length = 350 ÷ 20 = 17.5 cm
📝 Scale Factors
Scale Factor: The number you multiply by to enlarge or reduce a shape. A scale factor greater than 1 enlarges; between 0 and 1 reduces.
Example 3
A rectangle has sides 4 cm and 6 cm. It is enlarged by a scale factor of 3. What are the new dimensions?
Solution:
New sides = 4 × 3 = 12 cm and 6 × 3 = 18 cm
Example 4
A shape has been enlarged from a scale factor of 1 : 5. If the original side was 8 cm, what is the new length?
Solution:
New length = 8 × 5 = 40 cm
📝 Maps and Bearings
Map Skills: Use a ruler to measure distances on the map, then multiply by the scale to find real distances.
Example 5
On a 1 : 50000 map, two towns are 6.4 cm apart. Calculate the actual distance in kilometres.
Solution:
Actual distance = 6.4 × 50000 = 320000 cm
Convert: 320000 ÷ 100000 = 3.2 km
Quick method: 6.4 × 0.5 km = 3.2 km
Example 6
A road is 15 km long. How long will it appear on a 1 : 100000 map?
Solution:
Convert: 15 km = 1500000 cm
Map length = 1500000 ÷ 100000 = 15 cm
📝 Area and Scale
Area Scale Factor: If lengths are scaled by factor k, areas are scaled by factor k².
Example 7
A garden plan at 1 : 100 shows an area of 25 cm². What is the actual area?
Solution:
Length scale factor = 100
Area scale factor = 100² = 10000
Actual area = 25 × 10000 = 250000 cm² = 25 m²
🧮 Worked Examples — Exam Style
Scale drawings: real = drawing × scale factor, with UNITS converted. Bearings pair with scales for navigation questions (measure clockwise from north).
A plan scale is 1 : 200. A wall is 6.5 cm on the plan. How long is it really?
6.5 × 200 = 1300 cm = 13 m
M1 for × 200, A1 for 13 m (the cm → m conversion completes the mark).
On a map, town B lies on a bearing of 060° from A, 9 cm away. The scale is 1 : 50000. Find the real distance.
9 × 50000 = 450,000 cm = 4.5 km, on a bearing of 060°
M1 for the scale multiplication, A1 for 4.5 km. Bearings: always three figures, measured clockwise from north.
Exam wording to expect: scale as 1 : n or 1 cm : x m, "find the real distance", bearings from maps.
❓ Practice Questions
Q1: A map has scale 1 : 50000. A distance measures 8 cm. What is the real distance in km?
Q2: A drawing uses scale 1 : 40. A table is 1.8 m long in real life. How long is it on the drawing?
Q3: A shape with sides 3 cm and 5 cm is enlarged by scale factor 4. Find the new dimensions.
Q4: Two villages are 7.5 km apart. How far apart are they on a 1 : 25000 map?
Q5: A model car has scale 1 : 24. If the model is 18 cm long, how long is the real car?
Q6: A scale drawing uses 1 cm : 5 m. A room measures 4.5 cm by 3 cm on the plan. Find the real area.
Q7: A bearing of C from B is 240°. What is the bearing of B from C?
✅ Answers
- 4 km (8 × 0.5 km)
- 4.5 cm (180 ÷ 40)
- 12 cm and 20 cm
- 30 cm (750000 ÷ 25000)
- 432 cm or 4.32 m
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 8 × 50000 = 400,000 cm = 4 km
Q2. 180 ÷ 40 = 4.5 cm
Q3. 3 × 4 = 12 cm; 5 × 4 = 20 cm → 12 cm and 20 cm
Q4. 7.5 km = 750,000 cm → ÷ 25000 = 30 cm
Q5. 18 × 24 = 432 cm = 4.32 m
Q6. Real: 22.5 m × 15 m → area = 337.5 m²
Q7. Back bearing: 240 − 180 = 060°
📌 Key Facts to Memorise
- Real = drawing × scale factor; convert cm → m (÷100) or km (÷100,000)
- Area scales by the factor SQUARED — a 1 : 100 plan makes areas 1 : 10,000
- Bearings: three figures (045° not 45°), clockwise from north
- Back bearing: +180° if under 180, −180° if over
- Draw the north line at BOTH points for bearing problems
🗺️ In the Exam
Scales and bearings: 3–5 marks on Papers 2/3 (protractor needed for bearings). The unit conversion after scaling is where marks hide.
Links to: units (N13), ratio (R4), measuring (G15).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always check the units - convert to consistent units before calculating
- Remember: real = map × scale, map = real ÷ scale
- For area, square the scale factor
- For volume, cube the scale factor
- On maps, 1 : 50000 means 1 cm = 500 m = 0.5 km
🧠 Problem-Solving Strategies
Problem-Solving
Convert between map and real distances using the scale factor. Measure carefully with a ruler and protractor. Use proportional reasoning: if 1 cm = 50 m, then 3.4 cm = 3.4 × 50 = 170 m. For bearings, always measure clockwise from North.
Multi-Step Problem
A map has scale 1:25,000. Two towns are 14.4 cm apart on the map. A road is being built that will reduce the actual distance by 6 km. How far apart will the towns appear on a new map with the same scale?
Actual distance = 14.4 × 25,000 = 360,000 cm = 3.6 km. New actual distance = 3.6 − 6 = negative — this is impossible! Check: 14.4 × 25,000 = 360,000 cm = 3.6 km, so reducing by 6 km is not possible for this pair. The question needs checking. If the reduction were 2 km: new distance = 1.6 km = 160,000 cm. New map distance = 160,000 ÷ 25,000 = 6.4 cm.
⚠️ Common Errors
Watch Out!
1. Wrong: Scale 1:50,000 means 1 cm = 50,000 km Correct: 1 cm = 50,000 cm = 0.5 km
2. Wrong: Measuring bearings anti-clockwise from North Correct: Bearings are always measured clockwise from North
3. Wrong: Forgetting to convert map distance to actual distance before calculating Correct: Always multiply by the scale factor first
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A scale drawing of a field is shown with scale 1:2000. The field on the drawing measures 8.5 cm by 6.2 cm. A path of width 2 m runs diagonally across the actual field. Calculate the area of the field excluding the path. Give your answer in m².
Actual length = 8.5 × 2000 = 17,000 cm = 170 m. Actual width = 6.2 × 2000 = 12,400 cm = 124 m. Total field area = 170 × 124 = 21,080 m². Diagonal = √(170² + 124²) = √(28900 + 15376) = √44276 ≈ 210.4 m. Path area = 210.4 × 2 = 420.8 m². Field area excluding path = 21,080 − 420.8 = 20,659.2 m².
Mark scheme: M1 scale conversion, A1 actual dimensions, M1 area formula, A1 total area, M1 diagonal/path calculation, A1 final area
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A student draws two points A and B on a map (scale 1:10,000) and measures the distance as 7.2 cm. They then measure the bearing of B from A as 045°.
(a) Calculate the actual distance between A and B in kilometres.
(b) The student's ruler has a precision of ±0.1 cm. What is the range of possible actual distances?
(c) Is it possible to use scale drawings to find exact distances in real life? Explain your answer.
Answers: (a) 7.2 × 10,000 = 72,000 cm = 0.72 km. (b) 7.1–7.3 cm on map → 0.71–0.73 km actual. (c) No — measurements on drawings have limited precision, so real distances are always approximate.
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🎬 Video Resources
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