Geometry and Measurement · Measurement · Lesson 6
Bearings: Giving Direction with Three Digits
When a driver in Techiman asks the way to Kintampo, the answer is not 'go that way' but a direction everybody agrees on. Surveyors, pilots, ship captains and the Ghana Armed Forces all describe direction with a bearing. In this lesson you will learn to read, measure and draw bearings with a protractor.
By the end you will be able to
- ✔Explain that a true bearing is an angle measured clockwise from North.
- ✔Write any bearing as a three-digit number of degrees, e.g. 007° or 045°.
- ✔Use a protractor to measure the bearing of one point from another.
- ✔Draw a line of given length on a given bearing.
Key words
- Bearing
- The direction of one point from another, given as an angle measured clockwise from North.
- True bearing
- A bearing measured clockwise from the North line and written with three digits, from 000° to 360°.
- North line
- The straight line pointing North drawn through the point you are standing at (the starting point).
- Three-figure bearing
- A bearing written with three digits; 5° is written 005° and 60° is written 060°.
- Protractor
- A half-circle (or full-circle) instrument for measuring angles in degrees.
Curriculum reference
Content standard: B7.3.2.3 Demonstrate understanding of bearings, vector and its components using real life cases.
Indicator: Describe the bearing of a point from another point.
Starter
'Which way is Tamale from Kumasi?'
A trotro mate at Kejetia says, 'Tamale? Go north-ish.' A pilot cannot fly 'north-ish'. If a plane leaves Kotoka International Airport and the pilot is off by only 5°, after 500 km the plane is about 44 km away from where it should land! To avoid mistakes, everyone in the world uses the same system: measure the angle clockwise from North and write it with three digits. That angle is called the bearing.
1. North, the starting line
A direction needs two things: a starting point and a reference line. For bearings the reference line is always North. At the point you are standing, imagine an arrow pointing North. Every bearing is measured from that arrow.
The second rule is the direction of turning. We always turn clockwise, the way the hands of a clock move. A quarter turn clockwise from North takes you to East, a half turn to South, and three-quarters of a turn to West. A complete turn is 360°, which brings you back to North.
▶ Watch the four main directions appear. Each time, count the quarter-turns clockwise from North.
| Direction | Turn from North (clockwise) | Bearing |
|---|---|---|
| North (N) | no turn | 000° |
| North-East (NE) | half of a quarter turn | 045° |
| East (E) | quarter turn | 090° |
| South-East (SE) | 135° turn | 135° |
| South (S) | half turn | 180° |
| South-West (SW) | 225° turn | 225° |
| West (W) | three-quarter turn | 270° |
| North-West (NW) | 315° turn | 315° |
2. 'The bearing of B from A'
The words 'of B from A' tell you where to stand. You stand at A, the point after the word from, and look towards B. So 'the bearing of the school from the church' means: stand at the church, draw the North line there, and measure the angle to the school.
This order matters. The bearing of Kofi's house from Ama's house is not the same as the bearing of Ama's house from Kofi's. You will see in the next lesson exactly how they are related. For now, always ask yourself: where am I standing?
▶ Follow the three steps: North line at A, line to B, then measure clockwise from North.
The animation shows B on a bearing of 065° from A. B is a little north of East, which is exactly where 065° should be because 090° is due East.
3. Measuring a bearing with a protractor
To measure the bearing of B from A in a diagram, follow these steps carefully.
- 1Mark the point you are from (A). Draw a North line through A if one is not shown.
- 2Join A to B with a straight line.
- 3Place the centre of the protractor exactly on A, with its 0° line along the North line.
- 4Read the angle clockwise from North to the line AB. Use the scale that starts at 0° on the North line.
- 5Write the answer with three digits and the degree sign, e.g. 073°.
If AB lies to the right of the North line (the East side), the reading is between 0° and 180°, so you can read it straight off the protractor. If AB lies on the left (the West side), the angle you read from the North line going anticlockwise is less than 180°, but the bearing is 360° minus that angle.
| Where B lies | What the protractor shows | Bearing |
|---|---|---|
| East side of North line | Clockwise angle, e.g. 72° | 072° |
| West side of North line | Anticlockwise angle from North, e.g. 48° | 360° − 48° = 312° |
| Exactly West | 270° clockwise (or 90° anticlockwise) | 270° |
4. Drawing bearings and vectors
Drawing is the reverse of measuring. Suppose Esi leaves a junction A and walks 3 km on a bearing of 060°. We use a scale of 1 cm to 1 km, so the line is 3 cm long.
- 1Mark A and draw a North line upwards through it.
- 2Place the protractor on A with 0° on the North line and mark 060° on the East side.
- 3Draw a faint line from A through the mark, then measure 3 cm along it and label the end B.
- 4Check by measuring: the clockwise angle from North to AB should read 060°.
A journey with a distance and a bearing can be written as a pair, for example (3 km, 060°). The second example, (5 km, 120°), is drawn the same way: 5 cm at 120° clockwise from North, which points into the south-east area. Pairs like these are called vectors, and you will study them in the lessons that follow.
▶ Here is the bearing of 120° being drawn. Notice that B is below the East line, so it lies in the south-east.
Worked examples
Try each one yourself first, then open the solution and compare every step.
Example 1
Bearings of the main directions
Show the solution
- Step 1Start at North (000°) and turn clockwise.
- Step 2East is a quarter turn: 360° ÷ 4 = 90°, so 090°.
- Step 3South is a half turn: 360° ÷ 2 = 180°.
- Step 4West is three-quarters of a turn: 3 × 90° = 270°.
Answer: (a) 090° (b) 180° (c) 270°
Example 2
Writing a bearing with three digits
Show the solution
- Step 1The clockwise angle from North is 7°.
- Step 2A bearing needs three digits, so write two zeros in front of the 7: 007°.
Answer: 007°
Example 3
A bearing on the West side
Show the solution
- Step 1The angle 48° is measured anticlockwise from North.
- Step 2Bearings are clockwise, so subtract from a full turn: 360° − 48° = 312°.
- Step 3Check: 312° is between 270° and 360°, the north-west area. ✓
Answer: 312°
Example 4
Changing compass description into a bearing
Show the solution
- Step 1South is 180°.
- Step 2Turning from South towards East moves the direction back towards 090°, so subtract: 180° − 30° = 150°.
- Step 3Check: 150° is in the SE area (090° to 180°). ✓
Answer: 150°
Example 5
Drawing a vector from its bearing
▶ Draw the North line, turn 120° clockwise, then measure 5 cm along the line.
Show the solution
- Step 1Length: 5 km at 1 cm to 1 km gives 5 cm.
- Step 2Draw a North line at A, then turn 120° clockwise (that is 90° to East plus another 30° downwards).
- Step 3120° lies between 090° and 180°, so the line points South-East.
Answer: A 5 cm line pointing South-East (bearing 120°)
Summary
- A bearing is the angle measured clockwise from North, written with three digits (000° to 360°).
- 'The bearing of B from A' means stand at A and look at B, so the North line is drawn at A.
- Main bearings: N 000°, E 090°, S 180°, W 270°; NE 045°, SE 135°, SW 225°, NW 315°.
- If you read an angle anticlockwise from North, subtract it from 360° to get the bearing.
- A distance and a bearing, such as (3 km, 060°), describe a journey and form a vector.
Practice: 15 questions
5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.
You can practise without an account, but sign up free or sign in to save your progress.