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By-Hand Skills Sheet 5 — Bearings and Scale Drawing

Cambridge IGCSE Mathematics 0580 Extended · E4.3 · Name: ____________________   Date: __________
Bearings are measured and drawn with a protractor on paper — they cannot be practised by typing into a box. Every rule is here: three figures always, clockwise from north always, a north line at every point you measure from. Print this sheet, then work through it with protractor, ruler and compasses, exactly as in the exam.

In the print dialog choose A4 and Scale: 100% — not “Fit to page”. Then work through it with a sharp pencil, a ruler, a protractor, a pair of compasses and an eraser, exactly as you would in the exam. Check your answers on screen afterwards: the model answers below are hidden when you print, so the sheet comes out blank to work on.

What you need before you start. A protractor you can read to 1° (a full-circle one is easier for bearings past 180°, but the half-circle kind works with the 360-minus trick), a sharp HB pencil, a 30 cm ruler, a pair of compasses that holds its setting, and an eraser. No calculator anywhere on this sheet.
About the scale bars. Some diagrams carry a printed scale bar marked in kilometres. Set your compasses against that bar, not against a real centimetre ruler — if the printer has scaled the page even slightly, real centimetres will be wrong and the scale bar will still be right. In the exam the diagram is printed accurately and a real ruler is correct, so this is the one place this sheet differs from a real paper.
  1. Three figures, always. A bearing of 62° is written 062°. Two-figure bearings lose the mark.
  2. Clockwise from north, always. Never from south, never anticlockwise, never from the other point.
  3. North line at every point you measure from. “Bearing of B from A” = protractor on A, zero on A’s north line.
  4. Back bearing = bearing ± 180°, whichever keeps it between 000° and 360°.

Section 1  ·  Protractor warm-up

Three angles, three sizes. Decide acute / obtuse / reflex before you touch the protractor — that one habit catches almost every wrong-scale misread.
1.1 Measure the angle a with your protractor. Give your answer in degrees. [1]
Put the centre cross of the protractor exactly on the vertex, the zero line exactly along one arm — then read the scale that starts from 0 on that arm.
B a
How the marks are awarded B1 for 37°, accepted 35° to 39°. Reading the wrong scale gives 143° — a glance tells you the angle is clearly acute, so 143 cannot be right. Always sanity-check against "acute / obtuse / reflex" before writing anything down.
  • Marking points
  • The angle is 37° (accept 35–39°).
  • Centre cross on the vertex B, base line along the horizontal arm.
  • It is acute, so the answer must be under 90 — if you read 143° you used the wrong scale.
1.2 Measure the angle b. [1]
The angle is obtuse, so the answer is between 90° and 180° — decide that before you read the protractor.
C b
How the marks are awarded B1 for 124°, accepted 122° to 126°. If your protractor is too small to reach both arms, extend the arms with a ruler first — the angle does not change when the arms get longer.
  • Marking points
  • The angle is 124° (accept 122–126°).
  • Obtuse: between 90 and 180. Reading 56° means the wrong scale.
  • Extending an arm with a ruler never changes the angle — do it whenever the drawn arm is too short to reach the scale.
1.3 Measure the reflex angle c. [2]
No protractor reads past 360° and most stop at 180°: measure the small angle between the arms, then subtract from 360.
D c
How the marks are awarded M1 for measuring the acute angle between the arms (75°) and subtracting from 360. A1 for 285°, accepted 283° to 287°. Writing 75° when the arc marked is clearly the reflex one is the classic dropped mark.
  • Marking points
  • Acute angle between the arms: 75°. Reflex angle c = 360 − 75 = 285° (accept 283–287°).
  • The marked arc goes the long way round — that is what makes it reflex, and it must come out over 180.
  • This exact move is how you measure any bearing bigger than about 200° with a half-circle protractor.

Section 2  ·  Measuring bearings (E4.3)

Protractor on the “from” point, zero on its north line, read clockwise. Accepted tolerance throughout: ±2°.
2.1 Measure the bearing of B from A. [1]
"Of B from A" means the protractor sits on A. Zero on the north line, read clockwise round to AB.
N A B
How the marks are awarded B1 for 062°, accepted 060° to 064°. It must be written with three figures — 62° loses the mark on a real paper even when the measurement is perfect.
N A B 062°
  • Marking points
  • Protractor centre on A (the "from" point), zero along the north line.
  • Read clockwise from north to the line AB: 062° (accept 060–064°).
  • Three figures always: 062, never 62.
2.2 (a) Measure the bearing of Q from P.   (b) Measure the bearing of P from Q.   (c) Show by calculation how (a) and (b) are related. [3]
Part (b) needs the protractor on Q — and the angle goes past 180°, so use the reflex trick from 1.3.
N N P Q
How the marks are awarded B1 for (a) 118° (116–120). B1 for (b) 298° (296–300), measured at Q, clockwise from Q’s north line. B1 for (c): 118 + 180 = 298. The two north lines are parallel, so the two bearings always differ by exactly 180° — add 180 if the answer stays under 360, otherwise subtract 180.
N N P Q 118° 298° = 118° + 180°
  • Marking points
  • (a) At P: 118° (accept 116–120°).
  • (b) At Q, clockwise from north past 180: 298° (accept 296–300°). Measure 62° on the other side and do 360 − 62 if your protractor is a half circle.
  • (c) 298 = 118 + 180. Back bearing = bearing ± 180, whichever lands between 000 and 360.
  • Your two measured answers should differ by 180 — if they do not, one of them is misread.
2.3 Without measuring, write down the back bearing for each: (a) a ship sails on 062° — on what bearing does it return? (b) A is on a bearing of 210° from B — find the bearing of B from A. (c) A plane flies on 305° — on what bearing does it fly home? [3]
One rule: add 180° if the bearing is under 180°, subtract 180° if it is over. The answer must land between 000° and 360°.
How the marks are awarded B1 each: (a) 242° (b) 030° (c) 125°. All three figures every time — (b) is 030, not 30. No tolerance here: this part is arithmetic, not measurement.
  • Marking points
  • (a) 062 < 180, so add: 62 + 180 = 242°.
  • (b) 210 > 180, so subtract: 210 − 180 = 030° — and it must be written 030.
  • (c) 305 > 180, so subtract: 305 − 180 = 125°.
  • Adding 180 to 305 gives 485°, which is not a bearing — that is why the rule flips at 180.
2.4 Three ships are seen from the lighthouse L. Measure the bearing of each ship from L. [3]
Same centre point for all three. Ship Z is past 180° — reflex trick again.
N L ship X ship Y ship Z
How the marks are awarded B1 each: X 041° (039–043), Y 158° (156–160), Z 236° (234–238). Every answer clockwise from the same north line at L, every answer three figures.
N L ship X ship Y ship Z 041° 158° 236°
  • Marking points
  • X: 041° — and it is written 041, not 41.
  • Y: 158°, read straight off the clockwise scale.
  • Z: past 180, so measure the 124° on the anticlockwise side and do 360 − 124 = 236°.
  • All three from the one north line at L. Never re-zero the protractor on a ship.

Section 3  ·  Drawing bearings

Direction with the protractor, distance with compasses set against the scale bar. Construction lines stay in.
3.1 From O, draw and label rays on bearings of 075°, 160° and 310°. [3]
For 310°: either measure 310 clockwise in two steps, or measure 50° anticlockwise from north — because 360 − 310 = 50.
N O
How the marks are awarded B1 per ray, each within ±2° of the stated bearing, ruled from O and labelled. A ray on 050° instead of 310° scores nothing — anticlockwise is the single most common bearings error.
N O 075° 160° 310°
  • Marking points
  • 075°: protractor zero on north, mark at 75 clockwise, rule from O through the mark.
  • 160°: same, on the clockwise scale — the ray points down-right, south of east.
  • 310°: mark 50° on the anticlockwise side of north (360 − 310 = 50), or turn the protractor round past 180. Both land in the same place.
  • Label each ray with its bearing — an unlabelled ray can be matched to the wrong answer.
3.2 A boat B is 5 km from A on a bearing of 245°. Using the scale bar, mark the position of B. [2]
Direction first (protractor), then distance (compasses set against the scale bar, never a cm ruler).
N A 0 5 km
How the marks are awarded B1 for the direction: a ray from A within ±2° of 245°. B1 for the distance: B between 4.8 and 5.2 km from A measured against the scale bar. Both marks need the construction line left visible.
N A 0 5 km B 245° 5 km
  • Marking points
  • 245 > 180: measure 65° anticlockwise from south, or 245 clockwise in two steps — the ray points down-left.
  • Set the compasses to 5 km against the scale bar, arc across the ray, mark B where they cross.
  • Leave the ray and the arc in — they are the evidence for both marks.

Section 4  ·  Scale drawing — two-leg journeys

The exam favourite. Draw each leg to scale, then measure what the question asks. A fresh north line at every turning point.
4.1 A boat leaves the harbour H and sails 8 km on a bearing of 070°, then 5 km on a bearing of 155° to a buoy B. Draw the journey to scale. Measure the distance and the bearing of H from B — the direct course home. [4]
Draw a NEW north line at the turn: the second bearing is measured from north at the turning point, not at H.
N H 0 4 km
How the marks are awarded B1 first leg: 8 km within ±0.2 km, 070° within ±2°. B1 second leg from the end of the first: 5 km, 155°, same tolerances, measured from a north line drawn at the turn. B1 return distance: exact value 9.8 km, accepted 9.4 to 10.2 km read against the scale bar. B1 return bearing of H from B: exact value 280.6°, accepted 277° to 284°, measured at B.
N H 0 4 km N N turn B
  • Marking points
  • Leg 1: north line at H, 070° clockwise, 8 km against the scale bar.
  • Leg 2: fresh north line at the turn, 155° clockwise from it, 5 km. Reusing H’s protractor position bends the whole journey.
  • Join B back to H (dashed). Distance measures 9.8 km (accept 9.4–10.2); by calculation it is exactly 9.8 km.
  • Bearing of H from B, at B, clockwise from north: 280.6° (accept 277–284°). It is over 270 because home is west and slightly north.
  • Construction lines and all three north lines stay in.
4.2 From camp C a hiker walks 6 km on a bearing of 300°, then 7 km on a bearing of 210° to a waterfall F. Draw the journey to scale. How far, and on what bearing, must she walk to go straight back to C? [4]
Both bearings are over 180° — the 360-minus trick from 3.1 works for both. She walked 13 km; the straight line back is much shorter.
N C 0 4 km
How the marks are awarded B1 + B1 for the two legs (same tolerances as 4.1: ±0.2 km, ±2°, north line at the turn). B1 distance: exact value 9.22 km, accepted 8.8 to 9.6 km. B1 bearing of C from F: exact value 70.6°, accepted 068° to 074°. Note 13 km walked, 9.2 km back — if your straight line comes out longer than 13, something is drawn on the wrong side.
N C 0 4 km N N turn F
  • Marking points
  • Leg 1: 300° = 60° anticlockwise from north (up-left), 6 km.
  • Leg 2: fresh north line at the turn, 210° = 30° past south (down-left), 7 km.
  • Straight back: 9.2 km (accept 8.8–9.6; exact 9.22 km) on a bearing of 70.6° (accept 068–074°).
  • Sense check: the return bearing is the back bearing of roughly 250°, and 250 − 180 = 70. The two-leg drawing agrees with the ±180 rule.

Section 5  ·  Fixing a position

Two pieces of information cross at one point — two bearings, or a bearing and a distance. The crossing is the answer.
5.1 A ship S is on a bearing of 052° from station A and 318° from station B. Plot the position of S, then measure its distance from A and from B. [3]
One bearing gives a line the ship is somewhere on. Two bearings cross at exactly one point — that crossing IS the ship.
N N A B 0 4 km 10 km
How the marks are awarded B1 ray from A on 052° and ray from B on 318° (each ±2°), each drawn long enough to cross. B1 S marked at the crossing. B1 distances read against the scale bar: from A exact value 7.45 km, accepted 7.0 to 7.9 km; from B exact value 6.17 km, accepted 5.8 to 6.6 km.
N N A B 0 4 km 10 km S 052° 318°
  • Marking points
  • Ray from A on 052°, ray from B on 318° — each from its own north line.
  • They cross once. Mark S there; a position needs two bearings, one is never enough.
  • AS measures 7.4–7.5 km (exact 7.45), BS measures 6.2 km (exact 6.17), both set against the scale bar.
  • This is how coastguards actually fix a position — the exam question is the real procedure.
5.2 Radar shows a plane P is exactly 7 km from the tower T, and on a bearing of 108° from the beacon B. The plane is east of T. Plot P, then measure the bearing of P from T and the distance of P from B. [4]
"7 km from T" is a circle — compasses on T, radius set against the scale bar. The bearing ray from B cuts that circle twice; the words "east of T" choose which crossing is the plane.
N N B T 8 km 0 6 km
How the marks are awarded B1 circle centre T, radius 7 km set against the scale bar (±0.2 km). B1 ray from B on 108° (±2°). B1 P at the eastern crossing — the ray cuts the circle twice, and taking the wrong crossing loses this mark and both measurements. B1 both readings: bearing of P from T exact value 128.7°, accepted 126° to 132°; distance BP exact value 14.16 km, accepted 13.7 to 14.6 km.
N N B T 8 km 0 6 km P rejected — west of T 108° 7 km range circle
  • Marking points
  • Compasses on T, opened to 7 km against the scale bar: everywhere 7 km from T at once.
  • Ray from B on 108°. It enters the circle west of T and leaves it east — two crossings.
  • “East of T” picks the far crossing: that is P. Mark the rejected one so the examiner sees you chose deliberately.
  • Bearing of P from T: 128.7° exact (accept 126–132). Distance BP: 14.2 km exact 14.16 (accept 13.7–14.6).
  • Circle, ray and both crossings all stay visible.
  1. North lines first, every time. At the start, at every turn, at every station — before any angle is measured.
  2. Say the quadrant out loud before drawing. 070° is up-right, 155° down-right, 245° down-left, 310° up-left. A ray in the wrong quadrant is the most visible error on the page.
  3. Compasses on the scale bar for every distance — setting and checking, both.
  4. Check with the ±180 rule. Any return bearing you measure should be the outward answer ±180° when the path is straight — and roughly so even when it is not.
Cambridge IGCSE Mathematics 0580 Extended · by-hand skills sheet 5 · 13 exercises · E4.3 bearings and scale drawings · Print at 100% on A4 and complete in pencil.