Bearings: Edexcel GCSE Foundation maths revision notes

Bearings in the exam

A bearing is a direction, given as an angle. It tells you which way to go from one place to reach another, on a map or at sea.

In the exam you may be asked to write down a bearing, or to measure one with a protractor, for 1 or 2 marks. You may also be given a bearing and asked for the bearing back the other way, for 2 marks.

Harder questions use a scale drawing. You may have to draw a line on a bearing, or find a distance and a bearing by measuring, for up to 5 marks. Take a ruler and a protractor into the exam.

The three rules for a bearing

Measure from north. Start at the north line.
Turn clockwise, the way the hands of a clock turn.
Write the angle with three figures: 70^\circ is written 070^\circ, and 5^\circ is written 005^\circ

The word from tells you where to stand. For the bearing of B from A, stand at A, face north, then turn clockwise until you face B. The north line and the angle are both at A.

The compass points are easy bearings. North is 000^\circ, east is 090^\circ, south is 180^\circ and west is 270^\circ. Halfway between two of them is another 45^\circ round, so north-east is 045^\circ

Worked example 1: A compass direction as a bearing

A farm F is north-west of a village V. Write down the bearing of F from V.

Stand at V and face north. North-west is between west and north.
Turning clockwise, west is 270^\circ from north, and north-west is another 45^\circ round.
Your turn. Work out 270+45

Your answer

270+45=315
Answer: 315^\circ

Measuring and drawing a bearing

Put the centre of the protractor on the point you are measuring from.
Line up 0^\circ with the north line.
Count round clockwise from 0 to the line, on the scale that starts at 0 on the north line.
Write the answer with three figures.

A protractor only goes up to 180^\circ. For a bearing of more than 180^\circ, measure the angle the other way, from the line round to north, and subtract it from 360^\circ

Here the angle from AB round to north is 130^\circ, so the bearing of B from A is 360-130=230^\circ

To draw a bearing, use the same steps the other way round: mark the angle clockwise from the north line, then draw a line from the point through your mark.

Worked example 2: An angle measured the wrong way round

The diagram shows a pier P and a boat Q. The angle between the north line at P and the line PQ is 48^\circ. Work out the bearing of Q from P.

The 48^\circ goes anticlockwise from north, so it is not the bearing.
Turning clockwise from north to Q goes almost all the way round. It is a full turn, 360^\circ, take away the 48^\circ
Your turn. Work out 360-48

Your answer

360-48=312
Check: Q is north-west of P, and 312^\circ is between west, 270^\circ, and north, 360^\circ
Answer: 312^\circ

The bearing back the other way

If you know the bearing of B from A, you can work out the bearing of A from B. Going from B back to A is the opposite direction, which is half a turn, 180^\circ, different.

If the bearing is less than 180^\circ, add 180^\circ
If the bearing is more than 180^\circ, subtract 180^\circ

In the diagram, the bearing of B from A is 060^\circ, so the bearing of A from B is 60+180=240^\circ. If the bearing of D from C is 253^\circ, the bearing of C from D is 253-180=073^\circ

Never take the bearing away from 360^\circ to go back the other way. 360-60=300, and 300^\circ points up and to the left, not back towards A. A quick sketch with a north line at each point shows which answer makes sense.

Worked example 3: Finding the bearing back

The bearing of a mast M from a farmhouse F is 128^\circ. Work out the bearing of F from M.

Going from M back to F is the opposite direction, half a turn round.
128^\circ is less than 180^\circ, so add 180^\circ
Your turn. Work out 128+180

Your answer

128+180=308
Check: M is south-east of F, so F is north-west of M, and 308^\circ is between 270^\circ and 360^\circ
Answer: 308^\circ

Worked example 4: A bearing over 180 degrees, then back

The diagram shows a lighthouse L and a yacht Y. Kim puts her protractor on L and measures the angle from the line LY round to the north line. It is 104^\circ. (a) Work out the bearing of Y from L. (b) Work out the bearing of L from Y.

(a) The 104^\circ goes anticlockwise from north, so it is not the bearing. The bearing is a full turn take away the 104^\circ: 360-104=256^\circ
(b) Going from Y back to L is the opposite direction, half a turn different. 256^\circ is more than 180^\circ, so subtract 180^\circ
Your turn. Work out 256-180

Your answer

256-180=76, so the bearing is 076^\circ
Check: Y is between south and west of L, so L is between north and east of Y, and 076^\circ is between 000^\circ and 090^\circ
Answer: (a) 256^\circ (b) 076^\circ

Scale drawings with bearings

A scale drawing question gives a scale such as 1 cm represents 3 km. To mark a place, draw its bearing from the north line first, then measure its distance along that line with a ruler.

To find a real distance, measure the line between the two points in centimetres, then multiply by the scale. With 1 cm to 3 km, a line of 5.2 cm is 5.2\times3=15.6 km.

Sometimes the distance is hidden in a speed and a time. A boat that sails at 10 km/h for 3 hours goes 10\times3=30 km. Work out the real distance first, then divide by the scale to get the length on the drawing.

Map scales written as a ratio

Some maps give the scale as a ratio, such as 1:40\,000. This means 1 cm on the map stands for 40\,000 cm in real life. Measure the map length in centimetres and multiply by 40\,000

The answer is in centimetres, which is a huge number, so the question usually asks for kilometres. There are 100 cm in a metre and 1000 m in a kilometre, so 1 km is 100\,000 cm. To change centimetres into kilometres, divide by 100\,000

Worked example 5: A bearing and a distance on a map

The diagram shows a car park P and a viewpoint V on a map. The scale of the map is 1:40\,000. Find the bearing of V from P, and work out the real distance from P to V in kilometres.

Put the centre of the protractor on P, with 0^\circ on the north line. Counting round clockwise, the line PV passes 108^\circ
Measure the line PV with a ruler, not the north line. It is 6.5 cm long on the map.
Each 1 cm on the map is 40\,000 cm in real life: 6.5\times40\,000=260\,000 cm
Your turn. Change 260\,000 cm into kilometres.

Your answer

260\,000\div100\,000=2.6 km
Answer: the bearing is 108^\circ and the real distance is 2.6 km.

Worked example 6: Drawing a walk to scale

Sam walks from a car park C on a bearing of 125^\circ at 4 km/h for 3.5 hours, to a hut H. He makes a scale drawing using 1 cm to represent 2 km. Work out the length of CH on his drawing, and say how he draws it.

First find the real distance. Distance = speed \times time.
Your turn. Work out 4\times3.5

Your answer

4\times3.5=14 km
Each 1 cm on the drawing is 2 km, so divide by 2: 14\div2=7 cm
Draw a north line at C. With the protractor's centre on C and 0^\circ on the north line, mark 125^\circ clockwise.
Draw a line from C through the mark, 7 cm long, and label its end H.
Answer: CH is 7 cm long on the drawing, on a bearing of 125^\circ from C.

Common mistakes

Measuring anticlockwise from north. Bearings always turn clockwise; if the angle you know goes the other way, subtract it from 360^\circ.
Measuring from the wrong point. The bearing of B from A is measured at A, using the north line at A.
Taking the bearing from 360^\circ to go back the other way. Add or subtract 180^\circ instead, and check with a sketch.
Reading the wrong scale on the protractor. Use the scale that starts at 0 on the north line, and check whether the angle should be acute or obtuse.
Writing a bearing with fewer than three figures. Write 70^\circ as 070^\circ and 5^\circ as 005^\circ.
Measuring the wrong line or forgetting the scale. Measure the line between the two points, not a north line, then use the scale and give the units asked for.