Geometry: Scale drawings and bearings

What you'll learn

Scale drawings and bearings form a crucial component of the CIE IGCSE Mathematics Geometry syllabus, appearing regularly in both Paper 2 (Extended) and Paper 4 (Extended). This topic tests your ability to represent real-world distances accurately on paper, interpret maps and plans, and use the three-figure bearing system to describe directions. Expect questions worth 4-8 marks that combine measurement skills, geometric reasoning, and accurate construction techniques.

Key terms and definitions

Scale — the ratio between a distance on a drawing or map and the corresponding real-world distance, expressed as 1:n or in the form 1 cm represents x m.

Representative fraction — a scale written as a ratio without units, such as 1:50000, where both measurements are in the same units.

Bearing — a direction measured in degrees clockwise from north, always written as a three-figure number from 000° to 360°.

Back bearing — the bearing from point B to point A when you know the bearing from A to B, calculated by adding or subtracting 180°.

Scale factor — the multiplier used to convert measurements between the drawing and reality, where real distance = scale factor × drawing distance.

True north — the reference direction (000°/360°) from which all bearings are measured clockwise.

Plan view — a scale drawing showing a view from directly above, commonly used for floor plans and site maps.

Elevation — a scale drawing showing a side view of an object or building, often paired with plan views in technical drawings.

Core concepts

Understanding and working with scales

Scales convert real-world measurements to manageable drawing sizes and vice versa. In CIE IGCSE Mathematics examinations, you'll encounter scales in three formats:

Ratio format (1:n): A scale of 1:50 means 1 cm on the drawing represents 50 cm in reality. Both quantities must be in the same units.

Statement format: "1 cm represents 5 m" directly tells you the conversion. Convert to the same units before calculating: 1 cm represents 500 cm, giving a ratio of 1:500.

Fractional format: Sometimes written as 1/50000, equivalent to the ratio 1:50000.

Key calculations with scales:

• To find a real distance: multiply the drawing measurement by the scale factor
• To find a drawing distance: divide the real measurement by the scale factor
• To find the scale factor: divide the real distance by the corresponding drawing distance

When working with scales, maintain unit consistency. Convert all measurements to the same unit (usually centimetres or metres) before applying the scale factor. For map scales like 1:25000, note that 1 cm on the map represents 25000 cm (250 m) on the ground.

Creating accurate scale drawings

CIE IGCSE Mathematics examiners expect precise construction techniques when creating scale drawings:

Essential equipment:

• Sharp pencil (HB or H grade)
• Ruler marked in millimetres
• Protractor marked in single degrees
• Pair of compasses
• Eraser

Construction process:

1. Read the question carefully and identify the required scale
2. Convert all given measurements to drawing measurements using the scale
3. Draw a north line if bearings are involved
4. Mark a starting point clearly
5. Use ruler and protractor to measure angles accurately to the nearest degree
6. Draw lines of the correct scaled length
7. Label all points as instructed
8. Keep construction lines light but visible

For questions requiring measurements from your drawing, examiners typically allow ±2 mm tolerance for lengths and ±2° for angles. Show all construction lines unless specifically told to erase them.

Understanding the bearing system

Bearings provide unambiguous direction information using a standardized three-figure notation. Every bearing must be written with exactly three digits, measured clockwise from north.

Standard bearings:

• Due North: 000° (or 360°)
• Due East: 090°
• Due South: 180°
• Due West: 270°

Key principles:

Always measure bearings clockwise from north. A bearing of 045° represents northeast, while 315° represents northwest. The angle is always measured from the north line at the starting point, not the destination.

When describing bearings between two points, specify the direction clearly: "the bearing of B from A" means standing at point A and measuring the angle clockwise from north to point B.

Bearings on diagrams:

Mark north lines as vertical arrows pointing upward. If multiple points appear on a diagram, each point has its own north line running parallel to all others (north lines are always parallel).

Calculating back bearings

Back bearings allow you to find the return direction. If the bearing from A to B is θ:

• When θ < 180°: back bearing = θ + 180°
• When θ ≥ 180°: back bearing = θ - 180°

Example: If the bearing from Town A to Town B is 065°, the bearing from B back to A is 065° + 180° = 245°.

Example: If the bearing from Port X to Port Y is 310°, the bearing from Y back to X is 310° - 180° = 130°.

This relationship exists because you're measuring the opposite direction along the same straight line, which differs by 180°.

Combined problems with scale and bearings

CIE IGCSE Mathematics frequently combines scale drawings with bearings in multi-step problems. These questions test your ability to:

• Plot positions using given bearings and distances
• Measure bearings and distances from scale diagrams
• Calculate actual distances using the scale
• Find positions where multiple conditions intersect

Typical question structure:

1. Draw point A
2. From A, plot point B using a given bearing and distance
3. From B, plot point C using another bearing and distance
4. Measure the bearing and distance from A to C
5. Calculate the actual distance AC using the scale

Such questions often involve triangular routes (ship movements, aircraft navigation, hiking trails) and may require finding the shortest distance or optimal bearing.

Scale drawings in real-world contexts

Examination questions place scale drawings in practical scenarios:

Building and architecture: Floor plans showing room dimensions, requiring area calculations at actual size.

Navigation: Ship or aircraft movements between ports or airports, combining bearings with distance calculations.

Land surveying: Field boundaries, property maps, and site plans where accurate measurements matter.

Model making: Reducing large objects to scale models, or enlarging small designs to full size.

When answering these questions, read the context carefully to extract the mathematical information. Identify what the question provides (usually scale + some measurements) and what it asks you to find (typically a real distance, a bearing, or both).

Worked examples

Example 1: Basic scale drawing construction

Question: The diagram shows the position of point A. Using a scale of 1 cm to 20 m:

(a) Point B is 80 m from A on a bearing of 070°. Plot the position of B. [2]

(b) Point C is 60 m from B on a bearing of 160°. Plot the position of C. [2]

(c) Measure the distance AC on your diagram and calculate the actual distance from A to C. [2]

Solution:

(a)

• Convert 80 m to drawing scale: 80 ÷ 20 = 4 cm
• Draw north line at A
• Measure 070° clockwise from north
• Draw line 4 cm long to point B
• [1 mark for correct angle, 1 mark for correct distance]

(b)

• Convert 60 m to drawing scale: 60 ÷ 20 = 3 cm
• Draw north line at B (parallel to north at A)
• Measure 160° clockwise from north
• Draw line 3 cm long to point C
• [1 mark for correct angle, 1 mark for correct distance]

(c)

• Measure AC on diagram carefully (approximately 5.8 cm, accept 5.6-6.0 cm)
• Calculate actual distance: 5.8 × 20 = 116 m
• [1 mark for measurement, 1 mark for correct calculation with their measurement]

Example 2: Back bearing calculation

Question: A ship sails from Port P to Island I on a bearing of 138°.

(a) Calculate the bearing from Island I back to Port P. [2]

(b) The ship then sails from Island I to Port Q, a distance of 45 km on a bearing of 205°. Calculate the bearing from Q to I. [2]

Solution:

(a)

• The back bearing is found by subtracting 180° (since 138° < 180°)
• Back bearing = 138° + 180° = 318°
• [1 mark for method, 1 mark for correct answer]

(b)

• Since 205° > 180°, subtract 180°
• Bearing from Q to I = 205° - 180° = 025°
• [1 mark for method, 1 mark for correct answer with three figures]

Example 3: Scale interpretation and area

Question: A scale drawing of a rectangular garden uses a scale of 1:200.

(a) On the drawing, the garden measures 6 cm by 4.5 cm. Calculate the actual dimensions of the garden in metres. [2]

(b) Calculate the actual area of the garden in m². [2]

Solution:

(a)

• Scale factor = 200
• Actual length = 6 × 200 = 1200 cm = 12 m
• Actual width = 4.5 × 200 = 900 cm = 9 m
• [1 mark for correct method, 1 mark for both answers in metres]

(b)

• Area = 12 × 9 = 108 m²
• [1 mark for correct method, 1 mark for answer with units]

Note: A common error is to use the drawing measurements to find area (6 × 4.5 = 27 cm²) then scale it. While this works mathematically (27 × 200² = 1,080,000 cm² = 108 m²), it's more complex. Always scale the linear dimensions first.

Common mistakes and how to avoid them

• Forgetting to convert units before scaling — Students often multiply 1.5 m by scale factor 50 to get 75 m instead of converting to centimetres first (150 cm × scale becomes 3 cm on the drawing). Always work in the same units throughout; convert to centimetres for drawing measurements.

• Writing bearings with fewer than three figures — Writing 65° instead of 065° or 8° instead of 008° loses marks in CIE examinations. The three-figure rule is absolute; bearings must always show three digits, using leading zeros when necessary.

• Measuring bearings from the wrong north line — When finding the bearing of A from B, students sometimes measure from the north line at A instead of at B. The rule is simple: measure from the north line at the starting point (the point you're measuring "from").

• Confusing scale factor direction — Applying the scale factor upside down (dividing when you should multiply, or vice versa). Remember: to go from drawing to reality, multiply by the scale factor; to go from reality to drawing, divide by the scale factor.

• Incorrect back bearing calculations — Adding 180° when the bearing is already greater than 180° (e.g., 215° + 180° = 395°, which exceeds 360°). When the original bearing exceeds 180°, always subtract 180° to find the back bearing; when it's less than 180°, add 180°.

• Scaling area incorrectly — Multiplying a drawn area by the linear scale factor instead of the scale factor squared. If the scale is 1:50, areas scale by 50² = 2500, not by 50. Always scale linear measurements first, then calculate the area using the scaled dimensions.

Exam technique for Geometry: Scale drawings and bearings

• Command word recognition: "Draw" or "plot" questions require accurate construction with visible working (2-3 marks per point plotted). "Measure" questions expect you to use ruler or protractor on your own diagram (1-2 marks). "Calculate" questions want arithmetic using scale conversions (1-3 marks). "State" or "write down" questions about bearings need three-figure answers with the degree symbol.

• Show all construction lines clearly: Examiners award method marks even if your final answer has minor inaccuracies. Keep construction lines visible: north lines, bearing lines, and measurement marks. Use a sharp pencil and avoid making lines so faint they're invisible, or so heavy they obscure the diagram.

• Accuracy tolerances in construction questions: CIE mark schemes typically accept ±2° for angle measurements and ±2 mm for length measurements. When measuring from your own construction, examiners apply "error carried forward" – you get marks for correct method using your previous measurements, even if those measurements were slightly off.

• Always include units with final answers: Write "65 m" not just "65", and "245°" not just "245". Final answers without appropriate units may lose the accuracy mark even if the numerical value is correct. For bearings, the degree symbol is essential; for distances, specify whether metres, kilometres, or centimetres.

Quick revision summary

Scales convert between drawing and reality: multiply by scale factor for real measurements, divide for drawing measurements. Always use consistent units. Bearings are three-figure angles (000°-360°) measured clockwise from north; find back bearings by adding or subtracting 180°. For scale drawings, use sharp pencils, accurate protractors, and clear north lines. Plot points using bearing angles and scaled distances. Measure carefully with ±2° and ±2 mm tolerance. Check units on all final answers. Area scales by the square of the linear scale factor.