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Maths · Transformations and constructions

Scale drawings and bearings

Use map and drawing scales, measure and write three-figure bearings, find back bearings, and solve journey problems.

  • 6 key terms
  • All boards
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Warm-up

Answer each one, then check.

  1. 1

    How many centimetres are in a kilometre?

    Show answerHide answer

    100 000

  2. 2

    What do the angles round a point add up to?

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    \(360^\circ\)

  3. 3

    What do co-interior angles add up to?

    Show answerHide answer

    \(180^\circ\)

  4. 4

    Work out \(3.2 \times 50\,000\).

    Show answerHide answer

    160 000

  5. 5

    What is the compass direction opposite to north-east?

    Show answerHide answer

    South-west

Learning Objectives

  1. 1Use a scale in the form \(1 : n\) to change between drawing and real lengths.
  2. 2Measure and write three-figure bearings.
  3. 3Find the bearing back from B to A.
  4. 4Solve problems using bearings and right-angled triangles.

Map Scales

A scale of \(1 : 50\,000\) means 1 cm on the map is 50 000 cm in real life.

  • Map to real

    Multiply by the scale, then change units. 3.2 cm on a \(1 : 50\,000\) map is 160 000 cm, which is 1.6 km.

  • Real to map

    Divide by the scale. 4.5 km is 450 000 cm, so on a \(1 : 25\,000\) map it is \(450\,000 \div 25\,000 = 18\) cm.

  • Units

    100 000 cm = 1 km and 100 cm = 1 m.

BEARINGS

A bearing is an angle measured clockwise from north, always written with three figures.

North is 000°, east is 090°, south is 180° and west is 270°.

Measuring a Bearing

Follow the same steps every time.

  1. 1 Find the starting point

    The bearing of B from A starts at A.
  2. 2 Draw or imagine a north line at A

    It points straight up the page.
  3. 3 Measure clockwise

    From the north line round to the line AB.
  4. 4 Write three figures

    Add zeros at the front, e.g. \(065^\circ\).

A Back Bearing

The bearing of B from A is \(065^\circ\). Work out the bearing of A from B.

Show the solutionHide the solution
  1. 1 The north lines at A and B are parallel Co-interior angles add up to \(180^\circ\)
  2. 2 Add \(180^\circ\) to a bearing under \(180^\circ\) \(065 + 180\)
  3. 3 Work it out \(245\)

Answer\(245^\circ\)

A Journey with Bearings

A ship sails 8 km on a bearing of \(070^\circ\) and then 6 km on a bearing of \(160^\circ\). How far is it from its starting point?

Show the solutionHide the solution
  1. 1 The two bearings differ by \(160 - 70 = 90\), so the path turns through a right angle
  2. 2 Use Pythagoras \(8^2 + 6^2 = 64 + 36 = 100\)
  3. 3 Distance \(\sqrt{100} = 10\)

Answer10 km

Using a Map Scale

Two towns are 3.2 cm apart on a map with scale \(1 : 50\,000\). How far apart are they in real life?

Show the solutionHide the solution
  1. 1 Real length in cm \(3.2 \times 50\,000 = 160\,000\)
  2. 2 Change to kilometres \(160\,000 \div 100\,000\)

Answer1.6 km

Back Bearings

Add or subtract 180 so the answer stays between 0 and 360.

  • 065°

    Rule: + 180°. Bearing of A from B: 245°

  • 130°

    Rule: + 180°. Bearing of A from B: 310°

  • 250°

    Rule: − 180°. Bearing of A from B: 070°

  • 305°

    Rule: − 180°. Bearing of A from B: 125°

Treasure Trail

Start at a point marked A. Draw the route: 5 cm on a bearing of \(050^\circ\), then 4 cm on a bearing of \(140^\circ\), then 6 cm on a bearing of \(230^\circ\). Measure the final distance and bearing back to A. If the scale is 1 cm to 100 m, work out the real distance.

1. Draw a north line at each point.

2. Use a protractor from north.

3. Convert with the scale.

A good answer shows: Students should end near the start; the route back to A is roughly 3 to 4 cm on a bearing near 300°. Real distance is about 300 to 400 m.

Can I...?

  1. 1Use a map scale \(1 : n\).
  2. 2Convert between cm, m and km.
  3. 3Measure a bearing.
  4. 4Write a bearing with three figures.
  5. 5Find a back bearing.
  6. 6Draw a bearing accurately.
  7. 7Use bearings with Pythagoras.
  8. 8Explain why the north lines are parallel.

Summary & Exam Focus

  • Bearings are measured clockwise from north, with three figures.
  • Back bearing: \(\pm 180^\circ\).
  • Scale \(1 : n\): multiply to get the real length; divide to get the drawing length.
  • Draw a sketch and mark north lines.

Exam focus

The bearing of B from A is \(072^\circ\). Work out the bearing of A from B. (3 marks) (3 marks)

Add 180 when the bearing is less than 180, and subtract 180 when it is more. Draw a north line at both points and mark the co-interior angles.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Bearing
An angle measured clockwise from north, written with three figures.
Scale
The ratio of a length on a drawing to the real length.
Back bearing
The bearing of A from B when you know the bearing of B from A.
Clockwise
The direction the hands of a clock turn.
Three-figure bearing
A bearing written with hundreds, tens and units, such as 045°.
Compass point
One of N, E, S, W and their combinations.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 2 marks

    The scale on a map is \(1 : 50\,000\). Two towns are 3.2 cm apart on the map. Work out the real distance between the towns, in kilometres.

    Show answerHide answer

    Model answer

    \(3.2 \times 50\,000 = 160\,000\) cm \(= 1.6\) km.

    Mark scheme

    • \(3.2 \times 50\,000\) — M1
    • 1.6 — A1
  2. Question 2 Non-calculator 2 marks

    Two villages are 4.5 km apart. Work out the distance between them on a map with scale \(1 : 25\,000\). Give your answer in centimetres.

    Show answerHide answer

    Model answer

    \(4.5\) km \(= 450\,000\) cm. \(450\,000 \div 25\,000 = 18\) cm.

    Mark scheme

    • \(450\,000\) seen — M1
    • 18 — A1
  3. Question 3 Non-calculator 3 marks

    The diagram shows two points, A and B. The bearing of B from A is \(072^\circ\). Work out the bearing of A from B.

    Points A and B with north arrows, the bearing of B from A marked as 72 degrees, and the bearing of A from B to be found.
    Show answerHide answer

    Model answer

    \(72 + 180 = 252^\circ\). The north lines are parallel, so the co-interior angles add up to \(180^\circ\).

    Mark scheme

    • \(180 + 72\) or \(360 - (180 - 72)\) — M1
    • 252 — A1
    • A reason: parallel lines or back bearing — C1
  4. Question 4 Calculator 4 marks

    A ship sails 8 km on a bearing of \(070^\circ\). It then sails 6 km on a bearing of \(160^\circ\). Work out the distance of the ship from its starting point.

    Show answerHide answer

    Model answer

    The two bearings differ by \(160 - 70 = 90^\circ\), so the path forms a right angle. Distance \(= \sqrt{8^2 + 6^2} = \sqrt{100} = 10\) km.

    Mark scheme

    • Right angle found — M1
    • \(8^2 + 6^2\) — M1
    • 100 — A1
    • 10 — A1
  5. Question 5 Non-calculator 2 marks

    A lighthouse is due south-west of a harbour. Write down the bearing of the lighthouse from the harbour.

    Show answerHide answer

    Model answer

    \(225^\circ\).

    Mark scheme

    • 225 — B2
  6. Question 6 Non-calculator 2 marks

    The bearing of Q from P is \(250^\circ\). Work out the bearing of P from Q.

    Show answerHide answer

    Model answer

    \(250 - 180 = 070^\circ\).

    Mark scheme

    • \(250 - 180\) — M1
    • 070 — A1

Quick check

  1. What is the bearing of east?

    1. A000°
    2. B090°
    3. C180°
    4. D270°
    Show answerHide answer

    B: 090°

    East is a quarter turn clockwise from north: 090°.

  2. How should the bearing 45° be written?

    1. A45
    2. B4.5°
    3. C045°
    4. D450°
    Show answerHide answer

    C: 045°

    Bearings always have three figures.

  3. The bearing of B from A is 130°. What is the bearing of A from B?

    1. A310°
    2. B050°
    3. C230°
    4. D130°
    Show answerHide answer

    A: 310°

    \(130 + 180 = 310\).

  4. A map scale is 1 : 20 000. 5 cm on the map is how far in real life?

    1. A100 m
    2. B10 km
    3. C5 km
    4. D1 km
    Show answerHide answer

    D: 1 km

    \(5 \times 20\,000 = 100\,000\) cm = 1 km.

  5. Bearings are measured from...

    1. AEast, anticlockwise
    2. BNorth, clockwise
    3. CSouth, clockwise
    4. DWest, anticlockwise
    Show answerHide answer

    B: North, clockwise

    North, going clockwise.

  6. The bearing of B from A is 300°. What is the bearing of A from B?

    1. A300°
    2. B060°
    3. C120°
    4. D480°
    Show answerHide answer

    C: 120°

    \(300 - 180 = 120\).

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