Bearings, Read and Solved Step by Step

A bearing is a direction written as a three-figure angle, measured clockwise from north. Due east is 090, due south is 180, due west is 270. The three figures are not decoration; a bearing of 60 degrees is always written 060. Most bearings questions are really triangle questions wearing a compass, and the work is in reading the diagram correctly before any trigonometry begins.

Reading a bearing, and its back bearing

To read the bearing of B from A, stand at A, face north, and turn clockwise until you point at B. The angle you turned through is the bearing.

The back bearing, the bearing of A from B, differs by 180 degrees. If B is on a bearing of 060 from A, then A is on a bearing of 240 from B (060 + 180). If the forward bearing is more than 180, you subtract instead. Back bearings are how you find the angle inside the triangle at the middle point of a journey, which is the step most students miss.

Worked example, turning a journey into a triangle

A ship sails from A to B on a bearing of 060 for 5 km, then from B to C on a bearing of 150 for 4 km. Find the distance AC and the bearing of C from A.

First, the angle at B. The back bearing of A from B is 060 + 180 = 240. The bearing of C from B is 150. So the angle ABC, measured between BA and BC, is 240 - 150 = 90 degrees. The journey makes a right angle at B.

Because the angle at B is a right angle, AC follows from Pythagoras:

AC = √(52 + 42) = √(41) = 6.40 km to 3 significant figures.

For the bearing of C from A, find the angle BAC inside the triangle: tan(BAC) = 4/5, so BAC = 38.7 degrees. The bearing of B from A was 060, and C lies a further 38.7 degrees clockwise, so the bearing of C from A is 060 + 38.7 = 098.7 degrees.

The teaching point

A bearings problem is solved in two stages, and the first is not trigonometry. First read the bearings and the back bearings to find the actual angle inside the triangle; only then choose Pythagoras, the sine rule or the cosine rule to finish. The marks are lost at the reading stage, using a forward bearing where a back bearing was needed, far more often than in the trig. Draw the north line at each point, mark the clockwise angle, and the triangle reveals itself.

If your child can do the trigonometry but mis-reads the diagram, that is a precise, teachable gap. GPA's Secondary Math programme trains the read-first routine on real E-Math bearings questions. A short diagnostic consult will show where the reading goes wrong.

Every worked value on this page was checked independently before publishing.