Geometry and measures

Bearings and scale drawings

Q: Why are bearings written with three figures?

So they can't be misread: 045^\circ makes it clear the angle is under 100^\circ. It is the standard way navigators write directions.

9 past-paper questions, easiest first. Try each one, then reveal the answer.

Foundation & Higher8 of 9 from Foundation papers

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Revision guide by The Full Marks

Bearings and scale drawings: what you need to know

Bearings describe direction as an angle from north, and scale drawings let you turn measurements on a map into real distances. These questions test careful measuring with a ruler and protractor, so accuracy is everything.

Key facts

A bearing is measured from north, clockwise, and written with three figures: $045^\circ$ , not $45^\circ$ .
'The bearing of $B$ from $A$ ' means stand at $A$ , face north, and turn clockwise to face $B$ .
Back bearing: add $180^\circ$ if the bearing is under $180^\circ$ , otherwise subtract $180^\circ$ .
A scale of $1 : 50\,000$ means $1$ cm on the map is $50\,000$ cm in real life. A scale like ' $1$ cm represents $2$ km' works the same way: multiply map cm by $2$ to get km, or divide km by $2$ to get map cm.
$100$ cm $= 1$ m, and $100\,000$ cm $= 1$ km.

Measuring or drawing a bearing

Draw a north line at the point you are measuring from.
Put the protractor's centre on that point with $0^\circ$ on the north line.
Measure clockwise. If the angle is more than $180^\circ$ , measure the anticlockwise angle and subtract it from $360^\circ$ .
Write the answer with three figures.

Using a map scale

Measure the length on the map in cm.
Multiply by the scale number to get the real length in cm.
Convert to m or km.
To plot a point instead, work backwards: convert the real distance to map cm, draw the bearing from the north line, and mark the point that distance along it.

Worked examples

Example 1 Foundation · grade 3

A map of the Lake District has a scale of $1 : 50\,000$ . Two villages are $6.4$ cm apart on the map. How far apart are they in real life, in km?

Show the solution

Real distance $= 6.4 \times 50\,000 = 320\,000$ cm
$320\,000 \div 100\,000 = 3.2$ km

Example 2 Foundation · grade 5

The bearing of a lighthouse $L$ from a harbour $H$ is $070^\circ$ . Work out the bearing of $H$ from $L$ .

Show the solution

The two north lines are parallel, so the angle turns round by $180^\circ$ .
$070 + 180 = 250$
The bearing of $H$ from $L$ is $250^\circ$ .

Common mistakes

Measuring anticlockwise, or from the wrong point. Read 'from' carefully: that is where the north line goes.
Reading the wrong scale on the protractor. Ask yourself: is the angle acute or obtuse?
Converting cm to km by dividing by $1000$ instead of $100\,000$ .

Exam tips

Answers are only accepted within a small tolerance, so measure carefully with a sharp pencil.
Leave your north lines and construction marks on the diagram; they can earn method marks.
Some bearing questions are worked out, not measured. North lines are parallel, so use alternate angles or co-interior angles (adding to $180^\circ$ ) with the other angle facts in the diagram.

Questions people ask

How do you measure a bearing?

Draw a north line at the starting point, then measure the angle clockwise from north to the line joining the two points. Always give three figures, such as $075^\circ$ .

Why are bearings written with three figures?

So they can't be misread: $045^\circ$ makes it clear the angle is under $100^\circ$ . It is the standard way navigators write directions.

Exam questions: Bearings and scale drawings (9, easiest first)

Bearings and scale drawings3 marksGrade 1No calculatorFoundation

November 2024 · Paper 1 Foundation · Q7

Measure a side and an angle of an accurately drawn triangle and name the angle type.

Show answer

(a)	$9.5$ cm (9.3 to 9.7)	1 mark
(b)	acute	1 mark
(c)	$32°$ (30 to 34)	1 mark

Worked solution Checked against the mark scheme

(a) Place the ruler's zero on P and read off at Q: about $9.5$ cm.

(b) The angle is smaller than $90°$ , so it is acute.

Watch out: Reading the wrong scale on the protractor and giving 148°.

Bearings and scale drawings1 markGrade 1CalculatorFoundation

November 2021 · Paper 3 Foundation · Q5

Mark the point halfway along a given line segment.

Show answer

Answer	A cross at the midpoint of AB	1 mark

Worked solution Checked against the mark scheme

Measure the length of AB with a ruler.

Halve that length, measure that far from A along the line, and mark a cross there.

Watch out: Estimating by eye instead of measuring.

Bearings and scale drawings2 marksGrade 2CalculatorFoundation

November 2022 · Paper 2 Foundation · Q6

Use a ruler and protractor to measure one side and one angle of a drawn quadrilateral.

Show answer

(a)	4.5 cm (4.3 to 4.7)	1 mark
(b)	$110°$ (108 to 112)	1 mark

Worked solution Checked against the mark scheme

(a) Line the zero of the ruler up with A and read off at B: about 4.5 cm.

(b) The angle is obtuse, so read the larger scale on the protractor: about $110°$ .

Watch out: Reading the wrong scale on the protractor and giving about $70°$ .

Bearings and scale drawings2 marksGrade 3CalculatorFoundation

June 2023 · Paper 3 Foundation · Q17

Plot a point from a bearing and a distance using a map scale.

Show answer

Answer	Cross 5.5 cm from T on a bearing of 065°	2 marks

Worked solution Checked against the mark scheme

Scale: 55 km $\div$ 10 = 5.5 cm.

Put the protractor on T, measure 65° clockwise from North, and mark a cross 5.5 cm from T along that line.

Watch out: Measuring the angle anticlockwise from North.

Bearings and scale drawings4 marksGrade 3No calculatorFoundation

November 2020 · Paper 1 Foundation · Q13

Measure a bearing between two points on an accurate map, then use a map ratio to find the real distance in kilometres.

Show answer

(a)	$025°$ (accept 23 to 27)	1 mark
(b)	1.25 km (accept 1.2 to 1.3)	3 marks

Worked solution Checked against the mark scheme

(a) Put the protractor on the north line at A and measure clockwise to the line AB: about $25°$ , written $025°$ .

(b) AB measures about 5 cm on the map.

Real distance $= 5 \times 25\,000 = 125\,000$ cm $= 1250$ m $= 1.25$ km.

Watch out: Converting cm straight to km by dividing by 1000 instead of 100 000.

Bearings and scale drawings2 marksGrade 4CalculatorOn both tiers

November 2023 · Paper 2 Foundation · Q27

Find a back bearing from a given bearing.

Show answer

Answer	327°	2 marks

Worked solution Checked against the mark scheme

The return bearing differs by $180°$ .

$147 + 180 = 327°$

Watch out: Subtracting from 360 to give 213°.

Bearings and scale drawings4 marksGrade 4CalculatorFoundation

June 2022 · Paper 3 Foundation · Q13

Measure distances on a scale map to compare two routes, then measure a bearing.

Show answer

(a)	300 metres (210 to 390 accepted)	3 marks
(b)	288° (286 to 290 accepted)	1 mark

Worked solution Checked against the mark scheme

(a) Measure: AB ≈ 5 cm, BC ≈ 4 cm, AC ≈ 7 cm.

Parveen: $5 + 4 = 9$ cm, Susan: 7 cm, so a difference of 2 cm.

$2 \times 150 = 300$ metres

(b) At C, measure clockwise from North round to the line CA: about $288°$ .

Watch out: Measuring the bearing anticlockwise, or from A instead of C.

Bearings and scale drawings5 marksGrade 5No calculatorOn both tiers

November 2019 · Paper 1 Foundation · Q27

On a scale drawing, plot a boat's position from a bearing, speed and time, then measure its distance and bearing from a lighthouse.

Show answer

((i))	Distance in the range $20$ to $23$ km	4 marks
((ii))	Bearing in the range $317°$ to $330°$	1 mark

Worked solution Checked against the mark scheme

Distance $PQ = 12 \times 1.5 = 18$ km

On the drawing: $18 \div 4 = 4.5$ cm

From $P$ , measure $70°$ clockwise from north and mark $Q$ 4.5 cm along that line.

(i) Measure $QL$ in cm and multiply by 4 (about 20 to 23 km).

(ii) At $L$ , measure clockwise from north round to $Q$ : about $317°$ to $330°$ .

Watch out: Measuring the bearing anticlockwise from north.

Bearings and scale drawings3 marksGrade 5CalculatorHigher

Specimen · Paper 2 Higher · Q9

Use an isosceles triangle and a given bearing to find the bearing of one point from another.

Show answer

Answer	135°	3 marks

Worked solution Checked against the mark scheme

Triangle $ABC$ is isosceles with $AB = CB$ , so angles at $A$ and $C$ are equal.

$\angle BAC = (180 - 50) \div 2 = 65°$

Bearing of $C$ from $A$ $= 70 + 65 = 135°$

Watch out: Taking 65° as the bearing, forgetting to add the 70°.

Also tested in these questions

Their main topic is different, but part of each one tests this topic.

June 2024 · Paper 3 Foundation · Q6: Use a map scale to find a real distance and a map length.
June 2023 · Paper 2 Foundation · Q7: Measure a side and an angle of a drawn triangle, then name a triangle with two equal sides.
June 2019 · Paper 3 Higher · Q23: Use bearings, the cosine rule and the sine rule to find the bearing of one town from another.
June 2018 · Paper 3 Foundation · Q12: Measure a scale drawing, use a 1 : 200 scale, and find the real perimeter in metres.
Specimen · Paper 3 Foundation · Q6: Measure a side and an angle on an accurately drawn trapezium.