Year 9 Maths The Thai Guy

Lesson 3.3 · 30 min

Scales and Bearings on Maps

Using map scales and three-figure bearings together to plot journeys and find distances.

In this lesson you will learn to…

  • Measure and write a three-figure bearing
  • Convert between map distances and real distances using a scale
  • Make a scale drawing of a journey to find distances East/West and North/South
  • Find the back bearing of a journey

Why this matters

Pilots, sailors, hikers and search-and-rescue teams all navigate with bearings. Combined with a scale, a bearing turns “that way, a long way” into a precise instruction such as “sail 147∘147^\circ for 100 km” — and lets you work out exactly where you will end up.

Reading a bearing

DirectionBearing
North000∘000^\circ
North-East045∘045^\circ
East090∘090^\circ
South180∘180^\circ
South-West225∘225^\circ
West270∘270^\circ
North-West315∘315^\circ

Bearings between 000∘000^\circ and 090∘090^\circ point somewhere North-East; 090∘090^\circ–180∘180^\circ South-East; 180∘180^\circ–270∘270^\circ South-West; 270∘270^\circ–360∘360^\circ North-West.

Map scales

Map scale1 cm on the map is …
1:50 0001 : 50\,00050 00050\,000 cm =500= 500 m
1:100 0001 : 100\,000100 000100\,000 cm =1= 1 km
1:400 0001 : 400\,000400 000400\,000 cm =4= 4 km

To go map → real, multiply by the scale. To go real → map, divide.

Working backwards

If a ship ends 10 km East and 20 km South of port, draw the right-angled triangle to scale, join the port to the finishing point and measure the angle from North, clockwise. Here the line points South-East, a little East of due South, so the bearing is about 153∘153^\circ. The length of that line, converted with the scale, is the distance sailed (about 22 km).

Protractor tip. For bearings over 180∘180^\circ, measure the angle anticlockwise from North and subtract it from 360∘360^\circ — it is much easier than measuring round the long way.

Key rules and formulas

  • Three-figure bearings

    A bearing is an angle measured clockwise from North, always written with three figures: East is , South is , and is written .

  • Always start at the “from” point

    The bearing of B from A means: stand at A, face North, turn clockwise until you face B. Draw the North line at A.

  • Back bearings

    If B is on a bearing of from A, then A is on a bearing of from B. Add if the bearing is under , otherwise subtract it.

  • Scales

    A scale of means 1 cm on the map is cm km in real life. A scale like "1 cm to 20 km" works the same way.
Try it

Bearing explorer

Three-figure bearing
045°
Back bearing (the return trip)
225°
Distance East
70.7 km
Distance North
70.7 km

Drag the pointer, or use the slider. This one is set for a 100 km journey: watch how the East/West and North/South distances change as you turn. Try from the first worked example.

Worked examples

Worked example 1

A scale drawing of a journey

A ship sails from port on a bearing of for 100 km. Using 1 cm to represent 20 km, find how far East and how far South it has gone.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Finding a position from two bearings

Town C is on a bearing of from town A and on a bearing of from town B. How do you find C?

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Back bearing and a map scale

On a map, ship C is 7.5 cm from ship A. How far apart are they in real life? And if C is on a bearing of from A, what is the bearing of A from C?

Try it yourself first, then reveal the solution one step at a time.

Practice: scales and bearings

11 questions

Eleven questions. Answers you would get from a careful scale drawing are accepted — you do not need to be exact to the kilometre.
  1. Question 1 of 11Quick-fire drill
    Quick-fire: write each direction as a three-figure bearing, e.g. 045.

    8 questions · 45 seconds · answer as many as you can.

  2. Question 2 of 11Explore and match
    Set the pointer to a bearing of . Then read off roughly how far East and South a 100 km journey on that bearing goes.

    Move the controls until you match the question, then press Check.

    Three-figure bearing
    045°
    Back bearing (the return trip)
    225°
    Distance East
    70.7 km
    Distance North
    70.7 km
  3. Question 3 of 11Multiple choice
    Which of these is written correctly as a three-figure bearing?
  4. Question 4 of 11Type the answer
    B is on a bearing of from A. What is the bearing of A from B?
    °
  5. Question 5 of 11Explore and match
    Ship A is on a bearing of from ship B. Set the pointer to the bearing of B from A.

    Move the controls until you match the question, then press Check.

    Three-figure bearing
    045°
    Back bearing (the return trip)
    225°
  6. Question 6 of 11Type the answer
    A map has a scale of . Two parks are 3.6 cm apart on the map. How far apart are they in real life, in metres?
    m
  7. Question 7 of 11Type the answer
    On a map drawn at 1 cm to 20 km, how long should you draw a journey of 150 km?
    cm
  8. Question 8 of 11Complete the table
    A ship sails 100 km on each bearing. Use the bearing explorer or a scale drawing to fill in the distances, to the nearest km.
    BearingDistance East (km)Distance South (km)
    132°
    151°
    165°
  9. Question 9 of 11Step by stepChallenge
    Work backwards from the finishing point.
    After sailing from port P, a ship finishes 10 km East and 20 km South of its starting point.
    1. Roughly which direction did it sail?

  10. Question 10 of 11Choose all that apply
    A ship sails on a bearing of . Select every true statement.

    Select every correct answer — there may be more than one.

  11. Question 11 of 11Type the answer
    Ship A is 36 km from ship B. How far apart are they on a map with a scale of ?
    cm