Mathematics 0580 · syllabus C4.3, E4.3
Cambridge IGCSE Mathematics 0580 — Scale drawings and bearings topical past papers
Reading and making scale drawings and maps, converting with scales such as 1 : 50 000, and measuring and calculating three-figure bearings.
- Questions
- 40
- Years
- 2010–2026
- With mark scheme
- 93%
- Questions 2022–2026
- 25
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What Scale drawings and bearings covers
A scale drawing shows real lengths reduced by a fixed factor. A scale can be written in words, such as '1 cm represents 2 km', or as a ratio, such as 1 : 50 000, which means that 1 cm on the map is 50 000 cm, or 500 m, in real life. You change map distances into real ones by multiplying by the scale factor, and real distances into map ones by dividing, and you make scale drawings with a ruler and a protractor.
A bearing is a direction given as a three-figure angle measured clockwise from north, from 000° to 360°, so east is 090° and south-west is 225°. To measure the bearing of B from A, draw a north line at A and measure clockwise round to the line AB. The back bearing, the bearing of A from B, differs by 180°: if B is on a bearing of 070° from A, then A is on a bearing of 250° from B. Directions such as due east and due south are also used.
Scale drawings and bearings are Core content, so they can be set on all four papers. Trigonometry questions may use bearings, where a triangle formed by two journeys is solved with right-angled trigonometry or, on the Extended route, the sine and cosine rules, and map scales link to ratio and, for areas on a map, to the area factor of similar shapes.
In the syllabus
The headings this topic has in the Cambridge International 2025–2027 syllabus, as the board prints them.
- C4.3 Scale drawings
- E4.3 Scale drawings
What you should be able to do
- Use a scale in words or as a ratio to convert between map and real distances
- Make scale drawings with a ruler and protractor, and take measurements from them
- Measure and draw three-figure bearings clockwise from north
- Find a back bearing by adding or subtracting 180°
- Use angle facts with parallel north lines to calculate bearings
Key ideas
- A map scale of 1 : n means every real length is n times the length on the map
- A bearing is an angle turned clockwise from north, written with three figures, such as 045°
- Back bearing = bearing ± 180°, kept between 000° and 360°
- North lines at different points are parallel, so co-interior and alternate angles link the bearings at the two ends of a journey
How Scale drawings and bearings has been examined
From 2022 to 2026, Scale drawings and bearings was on Paper 3 · Calculator (Core) in 6 of 14 exam series, worth about 2% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Non-calculator (Core) | 4 | 1% | 3 of 14 |
| Paper 2 · Non-calculator (Extended) | 6 | 1% | 5 of 14 |
| Paper 3 · Calculator (Core) | 11 | 2% | 6 of 14 |
| Paper 4 · Calculator (Extended) | 4 | 1% | 3 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 3.
Filed in Momentum under: Map Scales.
A worked example
A ship sails from port P on a bearing of 060° to a buoy B. Find the bearing of P from B. On a map with scale 1 : 200 000, the distance PB measures 6.5 cm. Find the real distance in kilometres.
- The bearing of P from B is the back bearing: 060° + 180° = 240°.
- On the map, 1 cm represents 200 000 cm, so 6.5 cm represents 6.5 × 200 000 = 1 300 000 cm.
- Divide by 100 to get metres, 13 000 m, then by 1000 to get kilometres, 13 km.
Answer: Bearing 240°; distance 13 km
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Measuring a bearing anticlockwise, or from the wrong point: the bearing of B from A starts at A, from north, turning clockwise.
- Writing a bearing with two figures, such as 45°, instead of 045°.
- Converting units wrongly with a ratio scale: at 1 : 25 000, 4 cm on the map is 100 000 cm, which is 1 km, not 100 km.
- Using the length scale to convert an area on a map: areas scale by the square of the length factor.
- Adding 180° to a bearing that is already more than 180°: subtract instead, so the back bearing of 290° is 110°.
Exam technique
- Draw a north line at every point you measure from, and mark the bearing angle clockwise from it.
- Convert map distances in centimetres to real distances first, then change the units in a separate step.
- For bearing calculations, use the parallel north lines, with co-interior angles adding up to 180°.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.