Bearings: worked examples and method

Three figures

Bearings are written with three digits: 040°, not 40°.

North is 000°, east is 090°, south is 180°, west is 270°.

From the north line

Draw a north line at the point you are measuring from.

The bearing is the clockwise angle from that north line to the direction of travel.

The return bearing

The opposite direction differs by 180°.

If the bearing from A to B is 040°, the bearing from B to A is 220°.

Co-interior on north lines

North lines are parallel, so a bearing and an angle inside the triangle are linked by co-interior or alternate angles.

Do not assume the angle in the triangle is the bearing.

A diagram

Mark north, then the clockwise angle.

A sketch with the wrong direction of turn gives the wrong bearing.

Check the range

A bearing is from 000° up to but not including 360°.

If a calculation gives 400°, subtract 360°.

How a bearing is measured

Start by drawing a north line at the point you are standing at. The north line goes straight up the page. Put your pencil on that line and turn clockwise until you are facing the direction you want. The amount you turned is the bearing. Because the turn is from north, a small bearing such as 020^° is only a little east of north, while 160^° is almost south, just a little east of it.

Write the angle with three figures even when the number is less than 100. A turn of 8^° clockwise from north is the bearing 008^°, not 8^° and not 08^°. The leading zeros are part of the answer. Check the compass points as a sense check: if your direction is roughly north-east, the bearing should sit between 000^° and 090^°. If it is roughly south-west, it should sit between 180^° and 270^°.

Method: 1. Draw north, then measure the clockwise turn from that line. 2. Write three figures, so an 8^° turn is the bearing 008^°. 3. Keep north-east between 000^° and 090^°, and south-west between 180^° and 270^°.

Drawing a bearing

To draw the bearing of B from A, mark point A and draw its north line. Place a protractor so that 0^° sits on the north line and the scale increases clockwise. Mark the required angle, draw the ray, and label it with the three-figure bearing. If the diagram is a scale drawing, measure the distance from A along that ray using the scale, then mark B. The north line at A is only a construction line. Leave it on the diagram so a reader can see where the angle was measured from.

Reading a bearing is the same steps in reverse. Draw the north line at the point you are standing at, if it is not already there. Measure the clockwise angle from that north line to the path. Round only if the question asks you to. Then write three figures. A measured angle of 4^° is the bearing 004^°. A measured angle of 90^° is 090^°. Do not measure the smaller angle on the anticlockwise side and then hope it is the bearing. If you went the wrong way around the protractor, the answer will be 360^° minus the true bearing.

Method: 1. At the starting point, put 0^° on north and mark the clockwise angle. 2. Read a drawn path the same way, then write three figures. 3. A clockwise measure of 4^° is the bearing 004^°, and 90^° is 090^°.

Finding the back bearing

The path from A to B and the path from B to A are the same line travelled in opposite directions. Opposite directions differ by a half turn, which is 180^°. If the bearing of B from A is less than 180^°, add 180^°. If it is greater than 180^°, subtract 180^°. The bearing 070^° gives the back bearing 070^° + 180^° = 250^°. The bearing 217^° gives 217^° - 180^° = 037^°. Both answers still have three figures.

You can remember the rule as “add 180^°, then if the result is 360^° or more, subtract 360^°”. Adding 180^° to 250^° gives 430^°, and 430^° - 360^° = 070^°, which is the forward bearing again. The two north lines, one at A and one at B, are parallel because both point north. That is why a bearing and its back bearing also fit the angle facts for parallel lines: the angle the path makes with north at A and the angle it makes with the south direction at B are linked by that half turn.

Method: 1. Add 180^° when the bearing is below 180^°, and subtract 180^° when it is above. 2. The back bearing of 070^° is 070+180=250^°. 3. The back bearing of 217^° is 217-180=037^°.

The angle between two paths

Sometimes a journey changes direction. A walker goes from P to Q on one bearing, then leaves Q on a new bearing. The angle inside the journey is the angle between the two pieces of path at Q. To find it, first find the bearing looking back from Q to P. That back bearing and the new bearing are both measured clockwise from the same north line at Q, so their difference is the angle between those two directions.

Take the positive difference. If it is greater than 180^°, the angle inside the path is 360^° minus that difference, because the inside angle of a bend is the smaller one. A useful check is the turn of the heading. If the forward bearing changes from 64^° to 150^°, the walker turns through 150^° - 64^° = 86^°. The angle between the path just walked and the path ahead is the supplement, 180^° - 86^° = 94^°. The back-bearing method gives the same 94^°, which is the angle you mark on the diagram between the two paths.

Method: 1. At the bend, find the back bearing of the path you have just walked. 2. Subtract the two bearings measured from the same north line. 3. A heading change from 64^° to 150^° is a turn of 86^°, and the angle between the paths is 180-86=94^°.

Reading "the bearing of A from B"

Every bearing question hides one small phrase: "the bearing of A from B". The word after "from" tells you where to stand. Put your pencil on B, draw a north line through B, and then measure clockwise to the line B to A. If you stand at the wrong point you will measure a bearing that is 180 degrees out, which is the classic way to lose every mark on the question.

Real exam papers also test whether you can match a three figure bearing to a compass direction. A bearing of 045^° is north-east, 135^° is south-east, 225^° is south-west and 315^° is north-west. A statement such as "045^° means north-west" is a favourite one mark trap, so know the four diagonals by heart.

Method: 1. Find the word after "from" and mark that point as your starting point. 2. Draw a north line through the starting point, pointing straight up the page. 3. Measure or work out the angle clockwise from the north line to the line you need. 4. Write the answer with three figures, adding zeros at the front if it is under 100^°.

Tip: Check the answer against the compass. Is a bearing of 225^° pointing roughly down and to the left? If your sketch disagrees, you have started at the wrong point.

Bearings inside a triangle

Many exam questions give two or three bearings and ask for an angle or another bearing. The key fact is that north lines are parallel. Where a path turns at a point, you can use the co-interior angle rule (angles add to 180^°) or the alternate angle rule to move a bearing across to the next north line. Always add the north line at every vertex you use.

A reliable shortcut for the angle at a turning point Q is: angle = back bearing of the first leg - bearing of the second leg, taking the larger number first. For example, if you arrive at Q on a bearing of 072^°, then the bearing of P from Q is 252^°. Leaving on a bearing of 140^° makes the angle in the triangle 252 - 140 = 112^°. Check that your angle is inside the triangle by sketching it.

Method: 1. Draw a north line at every point where a bearing starts. 2. Work out each back bearing by adding or subtracting 180^°. 3. Subtract the two bearings that meet at the vertex to get the angle in the triangle. 4. Use the isosceles or angle sum facts to find any remaining angle, then add it to a bearing.

Tip: If the triangle is isosceles ("PQ = QR"), the two base angles are equal, so finish with (180 - apex)÷ 2.

Distances from bearings: Pythagoras and the cosine rule

A common higher tier question asks for a distance, such as "work out the direct distance from P to R" after a ship sails two legs. First find the angle at the turn using back bearings. If the angle is 90^° the legs make a right angled triangle and Pythagoras gives the third side. Otherwise, use the cosine rule a^2 = b^2 + c^2 - 2bccos A, with the angle between the two known sides.

To find a bearing at the end, work out the angle at the start with the sine rule or with tan if the triangle is right angled, then add or subtract it from a known bearing. State the bearing as three figures, and give any rounding the question asks for, such as "to the nearest degree" or "to 3 significant figures". Keep full calculator values until the last line.

Method: 1. Draw the two legs as a triangle and mark the known lengths. 2. Find the angle between the two known sides using back bearings. 3. Use Pythagoras for 90^° or the cosine rule for any other angle. 4. Use trigonometry again for a final bearing, then round only at the end.

Tip: Cosine rule on a calculator: work out 2bccos A as one block before subtracting from b^2 + c^2.

Explaining, comparing and checking bearings

Exam questions often ask "Is X correct? Give a reason for your answer" or "Why must Y be wrong?". For a one mark reason, link the number to a compass direction or to a fact about the diagram. For example, a ship sailing from D to E where E is to the right of D cannot be on a bearing greater than 180^°, because that points to the left of the page.

When you measure a bearing from a scale drawing, expect an accuracy of about 2^° either way and a length accurate to the nearest millimetre. Use the scale factor to turn the measured length into a real distance. A quick check is that the bearing from A to B and the bearing from B to A always differ by exactly 180^°, so a measured pair that does not do so means one of them was read from the wrong point.

Method: 1. Sketch the situation with a north line at the point the bearing is measured from. 2. Say which compass direction the bearing points to, using 090^°, 180^°, 270^° as anchors. 3. Compare with the claim and write a short reason that uses a number. 4. Check with the back bearing rule: the two bearings must differ by 180^°.