Cambridge IGCSE Mathematics 0580

Bearings

A bearing is a direction given as an angle measured from north, clockwise, written with three figures. Due east is 090°, due south is 180°, due west is 270°. All three parts of that definition are marked: Cambridge examiner reports have flatly stated that “very few candidates showed understanding of bearings” — and the failures are exactly these three rules, not the geometry.

Updated 20 August 2026

The three rules

  1. From north. Every bearing starts at the north line of the point you’re measuring at. “The bearing of B from A” is measured at A — the word after “from” tells you where to stand.
  2. Clockwise. Never anticlockwise, even when the target is west of you — a point to the north-west has a bearing around 315°, not 45°.
  3. Three figures. 72° is written 072°; due north is 000°. Two-figure answers lose the mark even when the angle is right.

Working habit that prevents most errors: at the point named after “from”, draw the north arrow first, then sweep clockwise to the target and mark the angle you actually need.

Back bearings

If the bearing of B from A is θ\theta, the bearing of A from B (the back bearing) is

θ+180∘ (if θ<180∘)θ−180∘ (if θ≥180∘)\theta + 180^\circ \ (\text{if } \theta < 180^\circ) \qquad \theta - 180^\circ \ (\text{if } \theta \geq 180^\circ)

Why it works: the two north lines are parallel, so the bearings are co-interior angles around the line AB. Questions love asking for the reverse direction as a one-mark follow-up — it should never cost time.

Bearings with the sine and cosine rules

Extended papers embed bearings inside triangle problems: a ship sails from A to B on one bearing, then B to C on another — find the distance AC or the bearing of C from A. The bearings themselves only do one job: they give you an angle inside the triangle. The recipe:

  1. Draw north lines at every vertex of the journey.
  2. Use parallel-line angle facts (co-interior angles sum to 180°) to convert the two bearings into the interior angle at the middle vertex.
  3. Apply the cosine rule for the unknown side — a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc\cos A — or the sine rule for an unknown angle.
  4. If asked for a final bearing, convert your triangle angle back: add or subtract it from a known bearing at that vertex, and give three figures.

Worked example

Worked example

A ship sails 8 km from P to Q on a bearing of 070°, then 5 km from Q to R on a bearing of 160°. Find the distance PR.

Step 1 — the interior angle at Q. The north line at Q is parallel to the north line at P. The bearing P→Q is 070°, so the back bearing Q→P is 070° + 180° = 250°. The bearing Q→R is 160°. The angle PQR between the two directions at Q is 250° − 160° = 90°.

Step 2 — with a right angle, Pythagoras.

PR=82+52=89≈9.43 kmPR = \sqrt{8^2 + 5^2} = \sqrt{89} \approx 9.43\text{ km}

The examiners’ construction here is typical: the two bearings were chosen so the interior angle comes out clean. If it hadn’t been 90°, step 2 becomes the cosine rule with the same two sides wrapped around the angle you found.

The mistakes that lose the marks

  1. 1.Measuring at the wrong point

    “Bearing of B from A” is measured standing at A. Circle the word after “from” in the question before drawing anything.

  2. 2.Two-figure answers

    072°, not 72°. It’s a notation mark and it is genuinely withheld — pad with leading zeros every single time.

  3. 3.Sweeping anticlockwise for westward targets

    A target west of north is in the 270°–360° range. If your answer for something “up and to the left” is under 90°, you went the wrong way round.

  4. 4.Using the raw bearing as the triangle's interior angle

    Bearings are measured from north; interior angles sit between journey legs. Convert first (co-interior angles, back bearings), then apply the sine or cosine rule.

Common questions

Why are bearings measured clockwise from north?

Convention inherited from navigation: compass cards put north at the top and number clockwise, so a single three-figure number describes any direction unambiguously — no “north-north-west-ish” needed.

What is the bearing of due west?

270°. The compass points in bearings: north 000°, east 090°, south 180°, west 270° — and the diagonals: north-east 045°, south-east 135°, south-west 225°, north-west 315°.

Are bearings Core or Extended?

Both tiers use three-figure bearings; the Extended papers are where bearings combine with the sine rule, cosine rule and area formulas inside multi-mark triangle problems.

Practise bearings against real mark schemes

Quanta has real Cambridge IGCSE Maths 0580 past-paper questions, broken into skill checkpoints, marked criterion by criterion the way examiners mark — and it tracks which skills you’re missing. Free for individual students.

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