G15Geometry & MeasuresFoundation & HigherGrades 3–7

Bearings

Q: Why do bearings use three figures?

So they cannot be misread or altered — "45" could become "145" by accident, but "045" is unambiguous. It is a navigation convention.

Q: Can a bearing be 360^?

Due north is written 000^; 360^ is the same direction but is not normally used.

Q: Are the sine and cosine rules needed for bearings?

Only at Higher, when the triangle formed has no right angle.

A bearing gives a direction as an angle measured clockwise from north, written with three figures: east is $09 0^{\circ}$ , south is $18 0^{\circ}$ , and a direction just past north-west is around $32 0^{\circ}$ . Bearings combine angle facts, scale drawings and, at Higher, trigonometry.

40 min AQA GCSE Maths

The big picture

Three rules define a bearing: measure from $north$ , turn $clockwise$ , and write $three figures$ . "The bearing of $B$ from $A$ " means stand at $A$ , face north, and turn clockwise until you face $B$ .

North lines at different points are parallel. That single fact turns every "find the bearing back" or "find the angle in the triangle" question into a parallel-lines problem — co-interior angles add up to $18 0^{\circ}$ .

What you'll learn

Measure and draw three-figure bearings
Find the bearing of $A$ from $B$ given the bearing of $B$ from $A$
Use bearings with scale drawings
Calculate distances and bearings using angle facts and trigonometry

Measuring and drawing bearings

Draw a north line at the starting point. Place your protractor on it and measure clockwise. For bearings over $18 0^{\circ}$ , measure the anticlockwise angle and subtract it from $36 0^{\circ}$ .

The bearing of

B

from

A

06 2^{\circ}

. The bearing of

A

from

B

06 2^{\circ} + 18 0^{\circ} = 24 2^{\circ}

1(a) Quarter turn clockwise from north:

09 0^{\circ}

2(b) South is

18 0^{\circ}

; south-west is

4 5^{\circ}

further:

22 5^{\circ}

3(c)

360 - 45 = 31 5^{\circ}

Answer(a)

09 0^{\circ}

(b)

22 5^{\circ}

(c)

31 5^{\circ}

Your turnSame method, new numbers — try it before moving on.

What is the three-figure bearing of south-east?

Your answer

Back bearings

If the bearing of $B$ from $A$ is $θ$ , the bearing of $A$ from $B$ differs by exactly $18 0^{\circ}$ : add $18 0^{\circ}$ if $θ < 18 0^{\circ}$ , otherwise subtract $18 0^{\circ}$ .

295 > 180

, so subtract:

295 - 180 = 115

Answer

11 5^{\circ}

Your turnSame method, new numbers — try it before moving on.

The bearing of $Q$ from $P$ is $04 7^{\circ}$ . Find the bearing of $P$ from $Q$ , in degrees.

Your answer

Why ±180°

The two north lines are parallel, and the path between the points is a transversal. The bearing at $A$ and the angle measured at $B$ back to the north line are co-interior angles — they add to $18 0^{\circ}$ — which is why the back bearing is always $18 0^{\circ}$ away.

Scale drawings

A scale such as $1$ cm $: 5$ km means every centimetre on the drawing is 5 km in real life. Draw the north line, measure the bearing, then measure the scaled distance along it.

6.5 \times 4 = 26

Answer

26

Your turnSame method, new numbers — try it before moving on.

A map has scale $1 : 50, 000$ . Two villages are 7 cm apart on the map. How far apart are they in real life, in km?

Your answer

Calculating with bearings

Sketch the situation with north lines at each point. Use angle facts (co-interior angles, angles on a straight line) to find angles inside the triangle, then use Pythagoras or trigonometry — or, at Higher, the sine and cosine rules.

1The angle at the start between east and the direct line:

tan^{- 1} (\frac{3}{5}) = 30.9 6^{\circ}

2Bearing is measured from north:

90 - 30.96 = 59.04

Answer

05 9^{\circ}

Your turnSame method, new numbers — try it before moving on.

A boat sails 8 km due south then 6 km due west. How far is it from its starting point, in km?

Your answer

Tip — Label each point, draw its north line, and mark every bearing you are given before calculating.

Think like an examiner

Always write three figures: " $4 5^{\circ}$ " loses the mark where " $04 5^{\circ}$ " gains it.
Accurate drawings are marked within $\pm 2^{\circ}$ and $\pm 2$ mm — use a sharp pencil.
Read the wording "from" carefully: the bearing of $B$ from $A$ starts at $A$ .

Bearing rules

from North, clockwise, 3 figures

e.g.

04 5^{\circ}

27 0^{\circ}

θ_{back} = θ \pm 18 0^{\circ}

Back bearing.

co-interior angles on north lines = 18 0^{\circ}

North lines are parallel.

Watch out for these

Measuring anticlockwise from north.

Bearings are always clockwise.

Measuring from the line joining the points instead of north.

Draw the north line at the starting point first.

Writing a bearing as

6 2^{\circ}

Use three figures:

06 2^{\circ}

Mixing up "the bearing of

A

from

B

" and "of

B

from

A

Start at the point after "from".

Stretch yourself

Ship $S$ is 12 km from port $P$ on a bearing of $07 0^{\circ}$ . Lighthouse $L$ is 12 km from $P$ on a bearing of $13 0^{\circ}$ . Find the distance $S L$ and the bearing of $L$ from $S$ .

Hint — Angle $S P L = 130 - 70 = 6 0^{\circ}$ , and $P S = P L$ .

Questions students ask

Why do bearings use three figures?

So they cannot be misread or altered — "45" could become "145" by accident, but "045" is unambiguous. It is a navigation convention.

Can a bearing be

36 0^{\circ}

Due north is written

00 0^{\circ}

;

36 0^{\circ}

is the same direction but is not normally used.

Are the sine and cosine rules needed for bearings?

Only at Higher, when the triangle formed has no right angle.

Key takeaways

Measure from north, clockwise, and write three figures.
Back bearing: add or subtract $18 0^{\circ}$ .
North lines are parallel — use co-interior and alternate angles.
Scale drawings: measure the bearing, then scale the distance.
Sketch, find angles inside the triangle, then use Pythagoras or trigonometry.

Exam-style questions

Practise it the way it is examined

3 original questions · 10 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 32 marks

(a)Write down the three-figure bearing of west.[1]
(b)The bearing of $B$ from $A$ is $10 8^{\circ}$ . Work out the bearing of $A$ from $B$ .[1]

Question 2Grade 54 marks

A plane flies 120 km due north from airport $X$ , then 50 km due east to $Y$ .

Work out the distance $X Y$ and the bearing of $Y$ from $X$ , to the nearest degree.[4]

Question 3Grade 74 marks

Town $B$ is 40 km from town $A$ on a bearing of $05 5^{\circ}$ . Town $C$ is 25 km from $B$ on a bearing of $14 5^{\circ}$ .

(a)Show that angle $A B C = 9 0^{\circ}$ .[2]
(b)Work out the distance $A C$ , to 3 significant figures.[2]

Unlimited practice

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Bearings

Measuring and drawing bearings

Back bearings

Scale drawings

Calculating with bearings

Practise it the way it is examined

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Related topics

Bearings

Measuring and drawing bearings

Back bearings

Scale drawings

Calculating with bearings

Practise it the way it is examined

Skill drill

Practise this topic

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