A map shrinks real distances by a fixed scale, and a bearing fixes a direction with three digits measured clockwise from north. Together they describe any journey on paper.
A bearing is an angle measured CLOCKWISE from NORTH, written with three figures (\(065^\circ\), not \(65^\circ\)). The bearing back the other way differs by exactly \(180^\circ\).
Fig. 1: the bearing of B from A is 065°; the bearing of A from B is 065° + 180° = 245°
Section 01
Drawing to scale.
In the lab, choose a scale and draw the plan of a room. Change the scale from 1 cm to 1 m to 1 cm to 2 m and see the drawing halve in size while the real room stays the same.
A scale like 1 cm to 2 km means every centimetre on the drawing stands for 2 km in real life.
A ratio scale like 1 : 25 000 means real lengths are 25 000 times the map length, in the SAME units: 1 cm on the map is 25 000 cm = 250 m.
Map to real: multiply by the scale factor. Real to map: divide by it.
Units: 100 cm = 1 m and 100 000 cm = 1 km. Convert at the end.
Accuracy: draw with a ruler, and measure lengths to the nearest millimetre and angles to the nearest degree.
Worked example Map distance to real distance
A map has scale 1 : 25 000. Two churches are 7.2 cm apart on the map. Find the real distance in kilometres.
Real distance \(= 7.2 \times 25\,000 = 180\,000\) cm.
\(100\,000\) cm = 1 km, so \(180\,000 \div 100\,000 = 1.8\) km.
Answer: 1.8 km
Worked example Real distance to map distance
On the same map, how long is a straight road of real length 3.5 km?
Convert to centimetres: \(3.5 \times 100\,000 = 350\,000\) cm.
Divide by the scale: \(350\,000 \div 25\,000 = 14\) cm.
Answer: 14 cm
Examiner trap
With a ratio scale both sides are in the SAME unit. 1 : 25 000 does not mean 1 cm to 25 000 m. Multiply first to get centimetres, then convert to metres or kilometres.
Section 02
Three-figure bearings.
In the bearings tab, drag point B around point A and read its bearing. Start with B due north (000°), then move it slowly clockwise through east (090°), south (180°) and west (270°). Then switch to the bearing of A from B and see it change by 180°.
Three rules: start at North, turn clockwise, write three figures (000° to 360°).
Compass points: N = 000°, NE = 045°, E = 090°, SE = 135°, S = 180°, SW = 225°, W = 270°, NW = 315°.
"Bearing of B from A": stand at A, face north, turn clockwise until you face B. The word "from" tells you where to stand.
Back bearing: the bearing of A from B is the bearing of B from A plus 180° (if that is under 360°), otherwise minus 180°.
Worked example Back bearings
(a) The bearing of B from A is \(065^\circ\) (Fig. 1). Find the bearing of A from B. (b) The bearing of Q from P is \(310^\circ\). Find the bearing of P from Q.
(a) The north lines at A and B are parallel, so the direction from B back to A is exactly opposite to the direction from A to B.
Opposite means half a turn, \(180^\circ\), round: \(065^\circ + 180^\circ = 245^\circ\).
(b) \(310^\circ + 180^\circ = 490^\circ\), which is more than 360°, so subtract instead: \(310^\circ - 180^\circ = 130^\circ\).
Answer: (a) \(245^\circ\) (b) \(130^\circ\)
Your turn
The bearing of a lighthouse L from a boat B is \(204^\circ\). Find the bearing of the boat from the lighthouse.
\(204^\circ\) is more than \(180^\circ\), so subtract: \(204^\circ - 180^\circ = 024^\circ\). Write it with three figures: \(024^\circ\).
Examiner trap
Read the question's word order carefully. 'The bearing of A from B' means stand at B. Measuring at the wrong point gives the back bearing, which is wrong by exactly \(180^\circ\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
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Bearings are measured
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What is the three-figure bearing of due east?
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A map scale is 1 cm to 5 km. Two towns are 3.4 cm apart on the map. How far apart are they?
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A plan is drawn to a scale of 1 : 200. A wall is 4.5 cm long on the plan. How long is the real wall?
Section 03
Journeys on a scale drawing.
Many questions combine both ideas: draw each leg of a journey to scale on the correct bearing, then measure the final distance and bearing from your drawing.
Worked example A two-leg journey
A boat sails 8 km due east from harbour H, then 6 km due south to point P. Using a scale of 1 cm to 1 km, make a scale drawing and find the distance and bearing of P from H.
Draw a north line at H. Due east is \(090^\circ\): draw 8 cm across to the right.
At that point draw a new north line; due south is \(180^\circ\): draw 6 cm straight down to P.
Measure HP: it is 10 cm, so the real distance is 10 km. (Pythagoras checks it: \(\sqrt{8^2 + 6^2} = 10\).)
Measure the angle clockwise from north at H to the line HP: about \(127^\circ\).
Answer: 10 km on a bearing of about \(127^\circ\)
Your turn
Point D is due west of point C. Write down the bearing of D from C and the bearing of C from D.
Due west is \(270^\circ\), so D from C is \(270^\circ\). The reverse direction is due east: \(090^\circ\).
Examiner trap
Draw a NEW north line at every turning point. Measuring the second bearing from the first leg's line, instead of from north, is the classic error.
Stage 2 · Exam
Exam-style questions
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The bearing of B from A is \(072^\circ\). Find the bearing of A from B.
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A map has a scale of 1 : 50 000. A path is 6.4 cm long on the map. Find its real length.
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Q is due south-west of P. Find the bearing of Q from P.
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A ship sails from harbour H on a bearing of \(140^\circ\) to a point S. On what bearing must it sail to return directly to H?
Skill unlocked
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