GCSE · Geometry and measures

Bearings

Ships, planes, hikers and search-and-rescue teams describe directions with a single number: the bearing. It's an angle measured clockwise from north, always written with three figures. On this page you'll read and draw bearings, turn them round (back bearings), use parallel north lines to find angles, make scale drawings, and finally calculate distances and bearings with trigonometry.

GCSE subject content G15: measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings. G20 and (Higher) G22: trigonometry, including the sine and cosine rules, applied to bearings problems.

1 · Three-figure bearings

A bearing is a direction given as an angle with three rules:

  1. measure from north,
  2. turn clockwise,
  3. write it with three figures: 45° becomes 045°, 7° becomes 007°.

So east is 090°, south is 180°, west is 270°, and north is 000° (or 360°). Drag the arrow round the compass. (Angles recap: Angle facts.)

2 · Measuring and drawing bearings

To measure the bearing of B from A: stand at A. Draw a north line at A. Put the centre of the protractor on A with 0° on the north line, and measure clockwise round to the line AB. If the angle is more than 180°, a 360° protractor helps; with a 180° protractor, measure the anticlockwise angle instead and subtract it from 360°.

3 · "From" matters: back bearings

The bearing of B from A and the bearing of A from B are different: they point opposite ways, so they differ by exactly 180°. This reverse direction is the back bearing.

back bearing = bearing + 180° (if the bearing is less than 180°)
back bearing = bearing − 180° (if the bearing is 180° or more)

Why? The north lines at A and B are parallel, and the line AB crosses both, so the two angles between AB and north on the same side are co-interior: they add to 180°. Drag A and B.

Bearing of B from A
Bearing of A from B

Read the question carefully: "the bearing of B from A" means you stand at A (the word after "from"). Underline it. Measuring at the wrong end gives an answer that's out by 180°.

4 · Bearings and angle facts

Exam questions often give two bearings and ask for an angle inside the triangle, or the other way round. Draw a north line at every point, then use angle facts: angles round a point add to 360°, angles on a straight line add to 180°, and with parallel north lines, alternate angles are equal and co-interior angles add to 180°. A ship sails from A to B, then turns and sails to C:

5 · Scale drawings with bearings

A scale drawing shrinks real distances by a fixed amount; the scale tells you how. Here 1 grid square (1 cm on the map) = 2 km. To plot a point: draw a north line, measure the bearing, then measure along the line the scaled distance (real distance ÷ 2 here). (Recap: Scale factors.) Drag the flag to the right place.

6 · Bearings with right-angled trigonometry

When a bearing makes a right-angled triangle with a north–south and an east–west line, SOH CAH TOA finds distances (see Trigonometry and exact values). Travelling d km on a bearing θ (for θ between 0° and 90°): the north line is adjacent to θ and the east line is opposite, so you go d cos θ north and d sin θ east. For other bearings, find the acute angle the route makes with the north–south line first.

7 · Bearings with the sine and cosine rules Higher

When the triangle has no right angle, use the cosine rule (two sides and the angle between) and the sine rule (see Sine and cosine rules). Two ships leave port P. Ship A sails on one bearing and ship B on another. The angle at P is the difference between the bearings; the cosine rule gives the distance between the ships.

Worked examples

B is directly south-west of A. What is the bearing of B from A?
South is 180°. South-west is halfway between south and west (270°): 180° + 45° = 225°.
The bearing of B from A is 070°. Find the bearing of A from B.
070° is less than 180°, so add 180°: 070° + 180° = 250°.
Check with co-interior angles: at B the angle from north clockwise to BA is 250°, and the anticlockwise angle is 360° − 250° = 110°; 110° + 70° = 180° ✓.
The bearing of Q from P is 300°. Find the bearing of P from Q.
300° is more than 180°, so subtract: 300° − 180° = 120°. (Adding would give 480°, which is more than a full turn.)
A ship sails from A to B on a bearing of 050°, then from B to C on a bearing of 140°. Find angle ABC.
Back bearing at B: the bearing of A from B is 050° + 180° = 230°.
At B, the ship's new direction is 140° and the direction back to A is 230°. The angle between them is 230° − 140° = 90°.
(Or: the co-interior angle at B between north and BA is 180° − 50° = 130°, measured anticlockwise; the turn to 140° clockwise leaves 360° − 130° − 140° = 90°.)
A ship sails 20 km on a bearing of 040°. How far east and how far north of its start is it?
The route is the hypotenuse; 40° is the angle with north. North is adjacent, east is opposite.
East = 20 sin 40° = 12.855… = 12.9 km. North = 20 cos 40° = 15.320… = 15.3 km (3 s.f.).
A walker goes 5 km east then 12 km north. Find her distance and bearing from the start, and the bearing she must walk to return.
Distance = √(5² + 12²) = √169 = 13 km.
Angle with north: tan θ = opposite ÷ adjacent = 5 ÷ 12, θ = 22.619…° → bearing 023° (nearest degree).
Return: back bearing = 023° + 180° = 203°.
Two ships leave port P. Ship A sails 8 km on a bearing of 030°; ship B sails 11 km on a bearing of 110°. How far apart are they, and what is the bearing of B from A? Higher
Angle APB = 110° − 30° = 80°.
Cosine rule: AB² = 8² + 11² − 2 × 8 × 11 × cos 80° = 185 − 30.561… = 154.438…, AB = 12.427… = 12.4 km.
Angle PAB (cosine rule for an angle): cos PAB = (8² + 12.427…² − 11²) ÷ (2 × 8 × 12.427…) = 0.4903…, PAB = 60.66…°.
Bearing: at A, the bearing back to P is 030° + 180° = 210°. B is anticlockwise from that direction by 60.66…°: 210° − 60.66…° = 149.3…° → 149°.

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