Geometry · GCSE Maths
Bearings
GCSE Maths bearings: three-figure bearings measured clockwise from North, back bearings, scale drawings, and angle rules with parallel lines and triangles.
Bearings are always three figures, clockwise from North. Draw North at the point, then measure clockwise. Back bearing differs by 180°.
The important bits
What you need to know
- 1
A bearing is a direction measured clockwise from North, written as three figures: 045°, 120°, 305°. Write 050° not 50°.
- 2
At a point, draw a vertical line pointing North (arrow up). Measure the angle clockwise from North to the direction of travel.
- 3
Back bearing (return direction): add or subtract 180°. From A to B bearing 060°, from B to A is 060° + 180° = 240°.
- 4
If the forward bearing is less than 180°, add 180° for the back bearing. If it is more than 180°, subtract 180°.
- 5
Scale drawings: pick a scale (e.g. 1 cm represents 5 km), draw North, plot bearings with a protractor, measure distances on the diagram.
- 6
Angle rules: angles on a straight line sum to 180°; angles at a point sum to 360°; alternate and corresponding angles with parallel North lines.
- 7
In navigation problems, often use sine rule or cosine rule after finding an angle from bearings — draw a triangle and label sides.
- 8
Always show North on every diagram at each point where a bearing is taken. Missing North loses method marks.
Quotations worth analysing
Short evidence. Real method.
“Bearings are measured clockwise from North”
North is 000° (or 360°). East is 090°. South is 180°. West is 270°. Anticlockwise is wrong for standard bearings.
“Back bearing = forward bearing ± 180°”
The return direction points the opposite way along the same line. Add 180° if forward < 180°; subtract if forward > 180°.
“Three figures always”
058° not 58°. Leading zero matters for consistency and shows you know the three-figure rule.
Go deeper
Drawing and reading bearings
Ship at P sails on bearing 125°. At P draw North.up. Place protractor with 0° on North and measure 125° clockwise. Mark the direction ray. If Q is on that ray 8 cm away on a scale 1 cm : 2 km, PQ represents 16 km. To find the bearing of P from Q, draw North at Q, measure clockwise from North to QP — that is the back bearing. Parallel North lines at P and Q help: the angle between QP and North at Q relates to angles in the triangle. Practice converting compass points: NE is 045°, SE is 135°. Examiners mix bearing questions with Pythagoras or trigonometry once the triangle is drawn.
Go deeper
Back bearings and collinear points
A to B bearing 037°. B to A: 037° + 180° = 217°. A to C bearing 290°. C to A: 290° − 180° = 110°. If two bearings from the same point differ by 180°, the points lie on a straight line in opposite directions. In a triangle PQR, if you know bearings of Q from P and R from P, the angle at P inside the triangle is the difference between those bearings (careful which way you subtract). Draw both rays from P, mark the angle between them — that is often the sine-rule angle. Show subtraction on the diagram: “angle P = 125° − 40° = 85°”.
Go deeper
Scale drawings and parallel North lines
Two ports: from A, B is on bearing 050°, distance 12 km. Scale 1 : 100 000 means 1 cm = 1 km (check the scale stated). Draw North at A, measure 50°, mark B 12 cm along. From B, C is bearing 140°. Draw North at B — parallel to North at A. The angle between BA (back bearing 230°) and BC (140°) can be found using angles at B. Alternate angles: North lines parallel, so angle between a ray and North at B equals corresponding angle at A in suitable layouts. After the diagram, use cosine rule for distance AC or sine rule for a missing angle. Always write the scale and show North on the sketch.
See the idea in action
Town B is on a bearing of 060° from town A. Find the bearing of A from B. Step 1: Draw North at A and B. Mark B on bearing 060° from A. Step 2: The line AB has direction 060° from A. Step 3: From B back to A is the opposite direction along the same line. Step 4: Back bearing = 060° + 180° = 240°. Answer: the bearing of A from B is 240°. Check: 240° is clockwise from North at B, pointing back toward A.
Exam technique
Turn knowledge into marks
Draw North at every point mentioned. Write bearings as three figures. For back bearings, add or subtract 180° and state which you did.
Common mistakes
Do not give these marks away
- 01
Measuring anticlockwise from North, or from East instead of North.
- 02
Writing two-figure bearings (58° instead of 058°) or forgetting to add 180° for the return direction.
- 03
Not drawing a fresh North line at each location, leading to wrong angles when two bearings meet at different points.
The bearing of B from A is 035°. What is the bearing of A from B?
A215°
B145°
C035°
D325°
Show the answer
215°. Back bearing = 035° + 180° = 215°. 145° might come from subtracting wrongly. 325° is 035° + 290° or other incorrect arithmetic.
Quick questions
If this is the bit you searched
Why must bearings have three figures?
Convention so 050° is not confused with 500°. Write 008°, 045°, 120° with leading zeros where needed.
How do I find a back bearing?
Add 180° if the forward bearing is less than 180°. Subtract 180° if it is greater than 180°. The result is the direction back along the same line.
Can I use bearings with trigonometry?
Yes. Draw the triangle, use bearing information to find angles inside the triangle, then apply sine rule, cosine rule or SOHCAHTOA.
What if North is not drawn on the diagram?
Add it yourself at each point. Bearings cannot be read or checked without a North reference at that point.