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Scale drawings and bearings

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Cambridge O Level Mathematics (Syllabus D) 4024

Scale drawings and bearings

The big idea: A scale tells you how to turn a drawing length into a real length, and a bearing tells you which way to face. Put them together and a map becomes a set of triangles you can measure or calculate.

Summary — What This Chapter Covers

Use a scale written as "1 cm to 5 m" or as a ratio such as 1 : 50 000.
Convert between map lengths and real lengths, taking care with units.
Bearings are measured clockwise from north and written with three figures, from 000° to 360°.
North, east, south and west are 000°, 090°, 180° and 270°; "due east" means on a bearing of 090°.
The bearing of A from B differs from the bearing of B from A by 180°.

1. Scales and Scale Drawings

Syllabus 4.3 · Paper 1 & 2

Two ways of writing a scale

"1 cm represents 5 m" means every centimetre on the drawing is 5 m in real life. A ratio scale 1 : 50 000 means every length in real life is 50 000 times the drawing length, in the same unit.

Converting real length = drawing length × scale number drawing length = real length ÷ scale number
1 m = 100 cm 1 km = 1000 m = 100 000 cm

Worked Example

A map has a scale of 1 : 25 000. (a) Two villages are 7.4 cm apart on the map. Find the real distance in kilometres. (b) A road is 3.2 km long. How long is it on a map with scale 1 : 40 000?

Practice Question 1

A room is 12 m long and 7.5 m wide. Draw it to a scale of 1 cm to 2 m: what are the dimensions on the drawing?

2. Interpreting a Scale Drawing

Syllabus 4.3 · Paper 1 & 2

Measure, then convert

To find a real distance from a scale drawing, measure the drawing with a ruler to the nearest millimetre and multiply by the scale. To find a real area from a map, remember that a 1 cm by 1 cm square on the map stands for a real square whose sides are each one scale length, so the area scale is the length scale squared.

Practice Question 2

On a map with scale 1 : 50 000, a lake covers 6 cm². Find its real area in km².

3. Three-figure Bearings

Syllabus 4.3 · Paper 1 & 2

Clockwise from north

To give the bearing of B from A: stand at A, face north, and turn clockwise until you face B. The angle turned is the bearing. Always write three figures: 65° is written 065°.

Direction	N	NE	E	SE	S	SW	W	NW
Bearing	000°	045°	090°	135°	180°	225°	270°	315°

"D is due east of C" means the bearing of D from C is 090°; "due south" is 180°.

Practice Question 3

Write as three-figure bearings: (a) west, (b) south-east, (c) an angle 20° clockwise from south.

4. Back Bearings

Syllabus 4.3 · Paper 1 & 2

From A to B, and back

The north lines at A and B are parallel, so the two bearings differ by exactly 180° (co-interior angles).

Back bearing If the bearing of B from A is less than 180°, add 180°. If it is 180° or more, subtract 180°.

Worked Example

The bearing of B from A is 025°. Find the bearing of A from B.

Practice Question 4

The bearing of Q from P is 250°. Find the bearing of P from Q.

5. Bearings Problems

Syllabus 4.3 · Paper 1 & 2

Angles inside the journey

Memorise

Ratio scale

Area on maps

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Scale drawings and bearings

Scale drawings and bearings

Two ways of writing a scale

Measure, then convert

Clockwise from north

From A to B, and back

Angles inside the journey

Ratio scale

Area on maps

Past papers that test Scale drawings and bearings

More Mathematics D topics