For the bearing of B from A, put your protractor at A. Draw a north line there and measure clockwise towards B. Write the result with three digits, such as 038° or 246°. The word “from” identifies the observation point; reversing the letters reverses the direction.
These bearings and scale drawings worked examples use invented points and distances. They describe small, flat-map geometry, with all north lines parallel. They are revision exercises, not navigation instructions for real journeys.
Make the north line part of every diagram
A bearing records direction using an angle measured clockwise from north. The compass directions provide four useful checks.
| Direction | Bearing | Position on a north-up page |
|---|---|---|
| North | 000° | Up |
| East | 090° | Right |
| South | 180° | Down |
| West | 270° | Left |
North is also sometimes written as 360°. This lesson uses 000° for that direction. Bearings between 000° and 090° point into the northeast quarter; between 090° and 180° they point southeast. Continue this quadrant check before accepting a calculated answer.
Draw a new north line at each point from which a bearing is measured. Do not reuse the original north arrow as the vertex for an angle at another point. The lines represent parallel directions, but their starting points differ.
Worked example 1: turn compass descriptions into bearings
A fictional marker B is 38° west of north from A. Find the bearing of B from A, then the reverse bearing.
The short angle from north towards the west is anticlockwise. A bearing follows the long clockwise turn instead: 360° − 38° = 322°. This points northwest, agreeing with the description.
For the reverse direction from B to A, rotate by half a turn. Since 322° is at least 180°, subtract 180°: 322° − 180° = 142°. Draw a parallel north line at B and imagine looking southeast towards A. The reverse bearing has the correct quadrant.
Now consider a different description: C is 24° east of south from D. South is 180° and moving towards east reduces that bearing. Therefore the bearing of C from D is 180° − 24° = 156°. “East of south” and “south of east” use different starting directions; read both words.
Back bearings without a reflex-angle guess
For a straight route on this flat-map model, the back bearing differs by 180°. Use one of two operations:
- If the forward bearing is below 180°, add 180°.
- If it is 180° or more, subtract 180°.
For example, a forward bearing of 287° has back bearing 107°. A forward bearing of 064° has back bearing 244°. Applying the rule twice returns the original bearing, which is a useful self-check.
Do not use 360° minus the forward bearing as a general reversal rule. For 064°, that gives 296°, a reflection across the north line rather than the opposite direction. A sketch shows the difference: the actual return from a northeast destination must point southwest.
Worked example 2: convert a scale before drawing
A map uses scale 1 : 150,000. A straight route is 7.2 km long. How long should it be on the map?
The scale compares lengths in the same units. One centimetre on the map represents 150,000 cm on the ground. Since 100,000 cm = 1 km, 1 cm represents 1.5 km. The required map length is 7.2 ÷ 1.5 = 4.8 cm.
Check in reverse: 4.8 × 1.5 = 7.2 km. If a drawn segment later measures 3.6 cm, its represented distance is 3.6 × 1.5 = 5.4 km. Drawing-to-ground conversion multiplies by the distance per centimetre; ground-to-drawing conversion divides by it.
A scale affects lengths but not bearings. A larger drawing of the same route keeps the same angles. Avoid measuring a diagram displayed on a phone or resized web page as though its physical centimetres were fixed. Construct your own drawing using the stated scale.
Worked example 3: construct and check a two-leg route
From P, a route goes 8 km on bearing 090° to Q, then 6 km on bearing 000° to R. Use 1 cm to represent 1 km. Find the direct distance from P to R and the bearing of R from P.
- Mark P and draw a north line vertically upwards.
- Measure clockwise 090° at P. Draw PQ 8 cm to the right.
- At Q, draw another north line parallel to the first.
- Draw QR 6 cm northwards. Mark its end R.
- Join P directly to R. Measure PR, then measure its clockwise angle from the north line at P.
| Point | East of P | North of P |
|---|---|---|
| P | 0 km | 0 km |
| Q | 8 km | 0 km |
| R | 8 km | 6 km |
Your drawing should give PR about 10 cm, representing 10 km, and a bearing about 053°. These are target values, not an official marking tolerance.
Check the distance independently: PR = √(8² + 6²) = √100 = 10 km. For the angle θ measured from north, tan θ = eastward distance ÷ northward distance = 8/6. Thus θ ≈ 53.13°, or 053° to the nearest degree.
The angle at P measured from east is about 36.87°. It is a correct triangle angle, but it is not the requested bearing. The route travelled totals 8 + 6 = 14 km, whereas the straight separation is 10 km. Decide which distance the question requests.
Worked example 4: north resets at the turning point
A second route travels 4 km from A to B on 030°, then 4 km from B to C on 150°. What is the direct bearing of C from A?
The first leg moves east by 4 sin 30° = 2 km and north by 4 cos 30° = 2√3 km. The second moves east by 4 sin 150° = 2 km and north by 4 cos 150° = −2√3 km. The northward and southward parts cancel.
C is therefore 4 km directly east of A: its bearing from A is 090°. A scale construction with two 4 cm legs should show the same result. Measuring 150° from the direction AB instead of from north at B would create a different route.
Three practice tasks with solutions
Practice 1: read the return direction
The bearing of Y from X is 073°. Find the bearing of X from Y.
Solution: 073° + 180° = 253°. Y is northeast of X, so X must be southwest of Y. That quadrant check supports the calculation. Label the final answer “X from Y” so the reversal is explicit.
Practice 2: combine a distance and a description
Z is 11 km from W, 27° south of west. State its bearing and the drawing length at 1 cm to 2 km.
Solution: West is 270°. Moving from west towards south reduces the bearing: 270° − 27° = 243°. The length is 11 ÷ 2 = 5.5 cm. Draw both from W, keeping angle and length calculations separate.
Practice 3: an independent right-angled route
A route goes 3 km east from L to M, then 4 km south from M to N. Find LN and the bearing of N from L.
Solution: LN = √(3² + 4²) = 5 km. N is southeast of L. The angle east of south is tan−1(3/4) ≈ 36.87°, so the clockwise bearing from north is 180° − 36.87° ≈ 143°. The back bearing is approximately 323°.
Frequently asked questions
Where does the protractor go?
At the point after the word “from”. Align its reference with that point’s north line, then measure clockwise towards the destination.
Why must I write three figures?
Three-digit notation makes the direction unambiguous: write 009° rather than 9°. A decimal bearing may retain three digits before the decimal point.
Can a bearing exceed 360°?
After a full turn the direction repeats. Reduce it to the usual single-turn range; for example, 405° represents the same direction as 045°.
Is the shortest drawn angle always the bearing?
No. Northwest bearings are reflex angles measured clockwise. A small angle west of north must be subtracted from 360°.
Should I measure or calculate?
Follow the question. For a scale drawing, show an accurate construction and measurement. Calculation can independently check it, but does not replace requested construction work.
Why can two careful drawings differ slightly?
Pencil thickness, protractor placement and ruler readings introduce measurement uncertainty. Keep construction lines clear and report sensible precision without inventing a marking allowance.
Do all north lines have to be parallel?
Yes, in these small flat-map school models. Large-distance Earth navigation has additional complications that these questions do not model.
Further practice and references
The existing Mathematics topical revision resource is login-gated. This lesson does not promise a particular file inside it. The bearing convention was checked against MathsGenie’s bearings guide; all routes and solved tasks here are original.
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