Secondary 2 G3 Mathematics Tutor Bukit Timah | Three-Figure Bearings and Scale Diagrams. Did you know? A bearing of 090° means exactly east, while 270° means west—and the reverse journey uses a bearing 180° different. A carefully drawn north arrow can prevent more errors than a complicated calculator formula.
This guide helps Bukit Timah Secondary 2 G3 Mathematics families interpret north, three-figure bearings, reverse bearings, straight-line scale maps, triangles and diagrams. It gives the quick rules first and then original worked examples. Some bearing and trigonometry applications may be later-year extensions in a particular school, so use your own chapter plan for immediate revision.
Bearings in these questions describe geometric directions of straight-line displacement, not real road navigation or driving-time predictions. The maps and journeys are fictional teaching diagrams. Direction, distance and time are separate quantities, and no route metrics are being claimed.
Find the part of this bearing guide you need
- Quick answer: where a bearing starts and which way it turns
- Four cardinal directions and four diagonal directions
- The three digits are not optional decoration
- Why every station needs its own north arrow
- A bearing is not simply the angle written beside a triangle
- Reverse bearings: one line, two points of view
- Bearing and map scale solve different pieces of a journey
- Using right triangles for non-cardinal directions
- What is measured on a sketch is not always measured on a road
- Thirty-six original bearings and scale clinics
- Reverse of 058 degrees
- Reverse of 275 degrees
- Reverse of 004 degrees
- Reverse of 185 degrees
- Reverse of 359 degrees
- A north-west angle
- Reverse map length
- A north-east displacement
- Four west and three north
- Bearing 030 with ten kilometres
- Bearing 060 with ten kilometres
- Bearing 120 with ten kilometres
- Bearing 045 and ten kilometres
- Bearing 090 followed by a scale length
- A no-calculator route to the first eight bearings
- A practical scale-and-bearing exercise
- A two-week study plan from diagrams to mixed problems
- Four types of bearings mistakes a tutor can repair
- Frequently asked: is a bearing the same as a compass direction?
- Frequently asked: when should I add 180 degrees?
- Frequently asked: can a scale tell us how long a journey takes?
- Frequently asked: why do students get an acute calculator angle but the wrong bearing?
- Frequently asked: do we need trigonometry for every bearings question?
- Frequently asked: what is a full-mark explanation?
- Bukit Timah Mathematics learning and official references
Quick answer: where a bearing starts and which way it turns
A three-figure bearing is an angle measured clockwise from north at the point of departure, normally written with three digits. North is 000°, east 090°, south 180° and west 270°. To find the reverse direction along the same straight line, add 180° when the original bearing is below 180°, or subtract 180° when it is 180° or above, then write the result as three digits.
A map scale of 1:n means 1 unit on the diagram corresponds to n of the same units in reality. For scale 1:10,000, one centimetre on paper represents 10,000 cm = 100 m straight-line distance. Keep the bearing measured at the correct point and the distances converted into consistent units.
Four cardinal directions and four diagonal directions
- 000° north: travel upwards on a north-oriented Cartesian sketch.
- 090° east: travel rightwards.
- 180° south: travel downwards.
- 270° west: travel leftwards.
- 045° north-east: halfway clockwise from north to east.
- 135° south-east: halfway clockwise from east to south.
- 225° south-west: halfway clockwise from south to west.
- 315° north-west: halfway clockwise from west back to north.
The three digits are not optional decoration
When a school question says ‘three-figure bearing’, express a five-degree direction as 005°, not 5°. Forty-five degrees becomes 045°. Three digits help readers distinguish 005°, 050° and 500°; the last is not a conventional one-turn bearing. The geometric angle is still five or forty-five degrees; the formatting clarifies direction.
A direction written as 360° is the same as north after one full clockwise revolution, but a standard three-figure bearing within the 000°–359° system is normally written 000°. If a worksheet specifies its own convention, follow that instruction.
Why every station needs its own north arrow
A bearing is measured at the starting point. Draw a vertical north line at A when measuring the direction from A to B. When finding the reverse direction from B to A, draw another north line at B. These north lines are parallel on the flat school diagram, but the angle is measured around a different point.
Students sometimes measure the reverse angle back at A or choose the nearest horizontal line as a reference. Those methods can produce a plausible-looking answer with the wrong origin. Circle the letters in ‘bearing of B from A’ to identify A as the point where the angle must be drawn.
A bearing is not simply the angle written beside a triangle
A right triangle may contain an interior angle of 40° while the corresponding three-figure bearing is 040°, 140°, 220° or 320°, depending on which quadrant the journey points into. Work out the compass quadrant first, then determine the clockwise turn from north. A calculator’s acute arctangent result alone does not supply the quadrant.
A useful check uses the target’s position relative to the starting point. If the point is to the west of the start, a bearing of 090° is impossible. If it is south-east, its bearing lies between 090° and 180°. These broad bounds often reveal an error before any detailed trigonometry.
Reverse bearings: one line, two points of view
If the bearing from A to B is 058°, then from B back to A it is 238°. Both describe the same line, viewed in opposite directions. Their angles differ by 180° modulo one full turn. This rule assumes a straight-line direction and the same north convention at both ends of a flat geometry diagram.
The reverse of 275° is 095°, and the reverse of 359° is 179°. The zero-padded writing is important. To check a reverse calculation, reverse it once more: adding or subtracting 180° again should return to the starting bearing modulo 360°.
Bearing and map scale solve different pieces of a journey
Suppose a diagram states that point B lies 4 cm from A on bearing 090°, with a scale of 1:25,000. The bearing tells us the eastward direction. The scale tells us that 4 cm represents 4×25,000=100,000 cm = 1,000 m. The result is a straight-line 1 km eastward displacement, not a road itinerary.
A complete school solution should name both quantities and preserve their independent units: 090° for direction and 1 km for distance. If the question asks for the length of a scaled line to draw, reverse the scale calculation by dividing actual length, in matching units, by the denominator.
Using right triangles for non-cardinal directions
If a displacement includes east and north components, draw a right triangle using those perpendicular directions. Pythagoras gives the straight-line length. To calculate a bearing in the north-east quadrant, an acute angle measured clockwise from north satisfies tan θ = east displacement ÷ north displacement, provided the north displacement is positive.
For 3 units east and 4 units north, the distance is 5 units and θ = tan⁻¹(3/4) ≈ 36.87°, so a bearing rounded to the nearest degree is 037°. But if the same magnitudes point south-east, the bearing is 180° − 36.87° ≈ 143°. Always position the displacement before choosing the final angle.
What is measured on a sketch is not always measured on a road
Geometric bearings assume a chosen north reference and straight connecting lines. A physical journey may follow bends, roads, slopes or obstacles. A 1 km east displacement need not imply a 1 km walk. This article is for school-scale diagrams and vectors only; it does not give travel routes, location tracking or safety navigation advice.
The mathematical model becomes useful when its assumptions are stated precisely. ‘Straight-line distance’ and ‘route distance’ are not synonyms. Parents and tutors should ask the pupil to say which of those is actually requested in the word problem.
Thirty-six original bearings and scale clinics
Eastward direction
Diagram or question. What is the three-figure bearing of a point due east of A?
Worked answer. 090°.
Why the direction or distance is correct. Bearings start at north and turn clockwise one quarter revolution. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. 009° would be almost due north, not east.
Independent check. Due west of A is 270°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Due north and due south
Diagram or question. Write the bearings of a point due north and one due south from A.
Worked answer. North is 000°; south is 180°.
Why the direction or distance is correct. Clockwise turns of zero and half a revolution describe those directions. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Do not use 090° for north because right on a page feels like the starting axis.
Independent check. East is 090° and west 270°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
North-east midpoint
Diagram or question. A point lies exactly north-east of A. Find its bearing.
Worked answer. 045°.
Why the direction or distance is correct. North-east is halfway between 000° and 090°. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Writing 450° adds a whole rotation and is not three-figure format.
Independent check. South-east is 135°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
South-west direction
Diagram or question. A point lies exactly south-west of A. Find its bearing.
Worked answer. 225°.
Why the direction or distance is correct. It is 45° further clockwise than south, or midway between south and west. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. The mirror direction 135° points south-east instead.
Independent check. North-west is 315°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Reverse of 058 degrees
Diagram or question. A to B has bearing 058°. Find B to A.
Worked answer. 058°+180°=238°.
Why the direction or distance is correct. Opposite directions along a line differ by half a turn. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Don’t subtract 58° from 180° without checking the intended reference.
Independent check. Reverse 238° again to get 058°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Reverse of 275 degrees
Diagram or question. A to B has bearing 275°. Find B to A.
Worked answer. 275°−180°=095°.
Why the direction or distance is correct. The original is greater than 180°, so subtract the half-turn. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Writing 95° instead of 095° loses the requested presentation.
Independent check. 095° back to A reverses to 275°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Reverse of 004 degrees
Diagram or question. An object is on bearing 004° from a station. Find the return bearing.
Worked answer. 004°+180°=184°.
Why the direction or distance is correct. The return points nearly south, four degrees past due south clockwise. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. 084° is not a half-turn from 004°.
Independent check. 184° reversed returns 004°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Reverse of 185 degrees
Diagram or question. One station is on bearing 185° from another. Find the return bearing.
Worked answer. 185°−180°=005°.
Why the direction or distance is correct. A direction just west of south returns just east of north. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Do not interpret 185° as a 185-degree interior angle.
Independent check. 005° reversed returns 185°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Reverse of 359 degrees
Diagram or question. A to B is 359°. Find the bearing from B to A.
Worked answer. 359°−180°=179°.
Why the direction or distance is correct. 359° points just west of north; its reverse points just east of south. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Writing 539° fails to reduce modulo a full turn.
Independent check. 179° reversed gives 359°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Small compass description
Diagram or question. The direction is north 30 degrees east. Write a three-figure bearing.
Worked answer. 030°.
Why the direction or distance is correct. Begin facing north and turn 30° clockwise toward east. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. 330° would describe north 30 degrees west.
Independent check. North 20 degrees west is 340°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
From south toward east
Diagram or question. The compass description is south 40 degrees east. Give bearing.
Worked answer. 180°−40°=140°.
Why the direction or distance is correct. From due south, rotate toward east, reducing the clockwise bearing. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. 220° describes south 40 degrees west.
Independent check. South 15 degrees west corresponds to 195°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
A north-west angle
Diagram or question. A line points 20° west of north. Find its bearing.
Worked answer. 360°−20°=340°.
Why the direction or distance is correct. It lies just before completing the full clockwise turn back to north. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. 020° points east of north instead.
Independent check. For 50° west of north, bearing becomes 310°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
An interior angle between bearings
Diagram or question. At one common starting point, rays have bearings 035° and 110°. Find smaller angle between them.
Worked answer. 110°−35°=75°.
Why the direction or distance is correct. Both rays use the same north reference and lie in an interval without wrapping across 000°. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Subtracting bearings from different origins would not generally give an included angle.
Independent check. Bearings 010° and 350° have smaller separation 20°, not 340°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Wrap across north
Diagram or question. Two rays from A point on bearings 340° and 020°. Find their smaller angle.
Worked answer. From 340° clockwise to 020° is 40°, the smaller separation.
Why the direction or distance is correct. The bearings straddle north, so direct difference 320° is the larger reflex gap. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. The smaller angle is not 320°.
Independent check. Try 350° and 010°: separation 20°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Map 1 to 25000
Diagram or question. A line on a 1:25,000 map measures 4 cm. Find actual straight-line length.
Worked answer. 4×25,000=100,000 cm=1000 m=1 km.
Why the direction or distance is correct. Scale ratio compares matching units before conversion. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Reporting 1000 km would ignore centimetres-to-metres.
Independent check. Two centimetres on this map represents 500 m. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Map 1 to 50000
Diagram or question. At scale 1:50,000, a line is 3.2 cm. Find real straight-line length.
Worked answer. 3.2×50,000=160,000 cm=1600 m=1.6 km.
Why the direction or distance is correct. A scale factor changes length before units are converted. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Do not multiply 3.2 by 50 and call the answer kilometres.
Independent check. Four centimetres would correspond to 2 km. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Map 1 to 10000
Diagram or question. A scale drawing at 1:10,000 shows 7.5 cm between A and B.
Worked answer. Actual straight-line distance=7.5×10,000 cm=75,000 cm=750 m.
Why the direction or distance is correct. One map centimetre equals 100 actual metres. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Converting 75,000 cm to 75 m is wrong by a factor of ten.
Independent check. At 5 cm, the true distance would be 500 m. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Map 1 to 200
Diagram or question. A model plan at 1:200 shows a 3 cm straight segment. Find real length.
Worked answer. 3×200 cm=600 cm=6 m.
Why the direction or distance is correct. The 200 factor is applied in centimetres, then converted. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. The plan does not imply walking distance or a travel-time estimate.
Independent check. A 4.5 cm segment corresponds to 9 m. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Reverse map length
Diagram or question. At scale 1:50,000, what drawn length corresponds to 2 km?
Worked answer. 2 km=200,000 cm; map length=200,000/50,000=4 cm.
Why the direction or distance is correct. Drawing length is actual length divided by scale denominator in matching units. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Multiplying when working backwards produces an impossibly huge line.
Independent check. A 1 km displacement would be 2 cm on the map. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Ten kilometres at 1 to 100000
Diagram or question. At 1:100,000, how long should a line representing 10 km be?
Worked answer. One centimetre represents 1 km, so draw 10 cm.
Why the direction or distance is correct. At this scale, 100,000 cm=1 km. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. A scale bar could be used as a secondary check.
Independent check. A 2.5 km line is 2.5 cm. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
A north-east displacement
Diagram or question. A travels 3 km east and 4 km north. Find straight-line distance and bearing from A.
Worked answer. Distance=√(3²+4²)=5 km; bearing=tan⁻¹(3/4)≈036.87°, or 037°.
Why the direction or distance is correct. Angle is measured from north toward east, not from the east axis. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Using tan⁻¹(4/3) gives the complementary angle measured from east.
Independent check. Reverse bearing from the end back to A is 217° rounded. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Six north then eight east
Diagram or question. A point lies 6 km north and 8 km east of start A. Find distance and bearing.
Worked answer. Distance=√(36+64)=10 km. θ=tan⁻¹(8/6)≈53.13°, so bearing 053°.
Why the direction or distance is correct. The horizontal and vertical displacements form a right triangle. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. The travel may happen along two legs, but straight-line separation is 10 km.
Independent check. Reverse bearing is 233° to the nearest degree. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
East and south
Diagram or question. The target lies 3 km east and 4 km south of A. Find bearing.
Worked answer. The acute angle from south toward east is tan⁻¹(3/4)=36.87°. Bearing=180°−36.87°≈143°.
Why the direction or distance is correct. The target is south-east, so the bearing must lie between 090° and 180°. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Simply writing 037° would point north-east.
Independent check. Straight-line distance is 5 km. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
West and south
Diagram or question. A target lies 3 km west and 4 km south. Find its bearing.
Worked answer. Angle west of south is tan⁻¹(3/4)=36.87°, so bearing=180°+36.87°≈217°.
Why the direction or distance is correct. South-west bearings lie between 180° and 270°. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Subtracting this acute angle would point south-east.
Independent check. Reverse to the starting point is about 037°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
West and north
Diagram or question. A target is 5 km west and 12 km north. Calculate bearing to nearest degree.
Worked answer. Angle west of north is tan⁻¹(5/12)≈22.62°. Bearing=360°−22.62°≈337°.
Why the direction or distance is correct. North-west directions approach 360° from below. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Writing 023° points east of north.
Independent check. Distance is 13 km using the 5-12-13 triangle. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Five east and twelve south
Diagram or question. A point lies 5 km east and 12 km south of the origin. Find bearing.
Worked answer. Angle east of south is tan⁻¹(5/12)≈22.62°, so bearing=180°−22.62°≈157°.
Why the direction or distance is correct. Its quadrant is south-east, constraining the answer between 090° and 180°. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Using 203° would put it south-west.
Independent check. Distance is 13 km. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Four west and three north
Diagram or question. A target lies 4 km west and 3 km north. What is the bearing?
Worked answer. Angle west of north=tan⁻¹(4/3)≈53.13°, so bearing=360°−53.13°≈307°.
Why the direction or distance is correct. North-west requires subtracting the acute angle from a full clockwise turn. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. An answer 053° would wrongly put the target east of north.
Independent check. Straight-line distance is 5 km. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Bearing 030 with ten kilometres
Diagram or question. A ray has bearing 030° and length 10 km. Find east and north components.
Worked answer. East=10sin30°=5 km; north=10cos30°=5√3≈8.66 km.
Why the direction or distance is correct. Bearing is measured from the north axis, so north uses cosine. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Swapping sine and cosine would make the ray closer to east than north.
Independent check. Components satisfy 5²+(5√3)²=100. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Bearing 060 with ten kilometres
Diagram or question. A ray bears 060° from north with length 10 km. Find its components.
Worked answer. East=10sin60°=5√3≈8.66 km; north=10cos60°=5 km.
Why the direction or distance is correct. The bearing is closer to east, so east displacement exceeds north. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Treating 60° as measured from east swaps the components.
Independent check. Pythagoras returns 10 km for the combined displacement. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Bearing 120 with ten kilometres
Diagram or question. A point lies 10 km away on bearing 120°. Describe components.
Worked answer. East=10sin120°=5√3≈8.66 km; north=10cos120°=−5 km, meaning 5 km south.
Why the direction or distance is correct. The ray lies in the south-east quadrant, so north component is negative. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Calling the north component +5 would place the point in the wrong quadrant.
Independent check. Its reverse bearing is 300°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Bearing 045 and ten kilometres
Diagram or question. A displacement is 10 km at 045°. Find north and east parts.
Worked answer. Both components are 10/√2=5√2≈7.07 km.
Why the direction or distance is correct. A 45° line splits the north-east right angle symmetrically. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Neither component is 10 km; 10 km is the diagonal.
Independent check. A 135° ray has equal east and south magnitudes. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Two legs of a journey
Diagram or question. A moves 3 km east to B, then 4 km north to C. Find A to C direct displacement.
Worked answer. Direct distance=5 km; bearing 037° from A to C rounded to a degree.
Why the direction or distance is correct. The net direction is measured from the original A, not from B. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Summing leg lengths to 7 km gives path length, not straight-line displacement.
Independent check. The bearing C to A is about 217°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Opposite horizontal legs
Diagram or question. A moves 6 km east then 2 km west. Describe the net displacement.
Worked answer. Net is 4 km east, bearing 090° from the start.
Why the direction or distance is correct. Opposite horizontal components subtract before finding direction. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Total travel of 8 km is not the net displacement.
Independent check. If second leg were 9 km west, net would be 3 km west on 270°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Opposite vertical legs
Diagram or question. A moves 5 km north and then 8 km south. Describe net displacement.
Worked answer. Net is 3 km south, bearing 180° from start.
Why the direction or distance is correct. North and south components have opposite signs. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. 13 km is path length, not net displacement.
Independent check. If second leg were only 3 km south, net would be 2 km north. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Compass notation south-west
Diagram or question. Translate south 15° west into a three-figure bearing.
Worked answer. Bearing=180°+15°=195°.
Why the direction or distance is correct. From south, rotating toward west increases the clockwise-from-north angle. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. 165° means south 15° east, a different ray.
Independent check. South 45° west is 225°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
True angle, not reverse bearing
Diagram or question. Two lines leave point A at 030° and 120°. Find the included angle at A.
Worked answer. The bearings differ by 90°, so the included angle is 90°.
Why the direction or distance is correct. Both angles share the same starting point and north reference. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Adding 180° would be a reverse-direction operation not required here.
Independent check. The two rays can form perpendicular sides of a right triangle. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Bearing 090 followed by a scale length
Diagram or question. A 1:50,000 map shows B 4 cm due east of A. State displacement.
Worked answer. Distance=4×500 m=2 km; bearing=090°.
Why the direction or distance is correct. The scale gives magnitude while the east compass direction gives angle. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Do not infer travel time or a drivable road from this straight line.
Independent check. On the same map a 3 cm westward displacement is 1.5 km at 270°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
Three-digit zero padding
Diagram or question. A ray is five degrees clockwise from north. How should its bearing be written?
Worked answer. Write 005°, not 5° or 050°.
Why the direction or distance is correct. Three-figure formatting makes the direction unambiguous. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. A 050° bearing is fifty degrees, ten times the intended turn.
Independent check. A two-degree bearing is 002°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
A wraparound return check
Diagram or question. A to B has bearing 190°. Find B to A and check.
Worked answer. Subtract 180°: reverse 010°. Add 180° to reverse 010° and recover 190°.
Why the direction or distance is correct. Two reverse operations should return the original directed line. A valid solution says where north is, which point is the origin and whether the value is an angle, straight-line length or component.
Mistake to diagnose. Writing 370° is an unnormalised direction outside usual bearing notation.
Independent check. A→B 090° corresponds to B→A 270°. Ask the pupil to redraw a small north arrow and verify that the answer lies in the intended compass quadrant.
A no-calculator route to the first eight bearings
Draw a compass rose and label north at the top. Mark the four cardinal directions 000°, 090°, 180° and 270°. Then bisect each quadrant to obtain 045°, 135°, 225° and 315°. Once these eight anchors are secure, bearings such as 030°, 140° or 340° become easier to place by comparing with nearby known directions.
This is a valuable prerequisite before trigonometry. The pupil should be able to reject a bearing in the wrong quadrant using only the drawing. If the point lies north-west, a result of 050° cannot be correct even if a calculator displays an acute angle of fifty degrees.
A practical scale-and-bearing exercise
On squared paper, draw A at the centre and a north arrow through it. Use a fictional scale of 1 cm to 100 m. Draw B 4 cm directly east of A and C 3 cm directly north of B. Now calculate the displacement from A to C. It is 400 m east and 300 m north, with a direct separation of 500 m by Pythagoras.
Its bearing from A is tan⁻¹(400/300) ≈53.13°, or 053° to the nearest degree. It is not enough to write ‘north-east’ if the question requests three figures. For the return direction, add 180° to obtain 233°. The geometric model says nothing about whether a real road can follow either straight line.
A two-week study plan from diagrams to mixed problems
First week: Master compass directions, three-figure notation and reverse bearings. Practice drawing a new north arrow at each stated starting point. Convert one or two map distances between centimetres and metres each day. Explain why the direction from A to B and B to A differ, even though the segment has the same length.
Second week: Add simple north/east right triangles. Begin with 3-4-5 and 5-12-13 distances before calculating acute bearings. Then introduce south-east and north-west quadrants where an acute calculator angle needs an addition or subtraction from 180° or 360°. Finish with one mixed scale-map problem and a check of the reverse direction.
Four types of bearings mistakes a tutor can repair
A reference error begins measuring from east rather than north. An origin error puts the north arrow at the wrong named point. A quadrant error uses an acute arctangent result without adjusting its compass orientation. A scale error mixes diagram centimetres with actual metres or kilometres. They can produce similar-looking final answers but require different teaching corrections.
The tutor should ask the pupil to point to the origin and the north line before reviewing the algebra. Once the sketch is correct, verify the units and the arithmetic. A student who can classify an error is more likely to avoid repeating it in a changed question.
Frequently asked: is a bearing the same as a compass direction?
A bearing is a precise numerical way of recording direction clockwise from north. ‘North-east’ describes a broad familiar direction, while 045° specifies exactly halfway between north and east in the flat model. A bearing such as 037° is still north-east, but not the exact 45° diagonal. The requested precision depends on the problem.
Frequently asked: when should I add 180 degrees?
Add or subtract 180° when reversing the direction along the same straight line, so the origin and destination swap. Do not automatically add 180° when moving between two legs of a triangle; their included angle depends on the actual bearings measured at the relevant vertex. Naming A, B and C carefully prevents this confusion.
Frequently asked: can a scale tell us how long a journey takes?
No. A geometric scale relates diagram lengths to straight-line actual distances. Travel time requires a speed or a real route and cannot be invented from a compass angle. A 1 km separation may correspond to a different walking or driving distance. This is a Mathematics drawing guide, not a routing service.
Frequently asked: why do students get an acute calculator angle but the wrong bearing?
Inverse tangent usually returns an acute angle determined by a ratio of side magnitudes. Without considering whether the target is north-east, south-east, south-west or north-west, a pupil may place that acute angle in the wrong quadrant. Draw the point first, decide the broad bearing range, and only then adjust the calculator angle.
Frequently asked: do we need trigonometry for every bearings question?
No. Cardinal and diagonal bearings, reverse bearings and straightforward map scales can be answered without trigonometry. When component lengths form an unfamiliar right triangle, Pythagoras and trigonometric ratios may become useful. Schools introduce these topics in different sequences, so build the compass and scaling foundations first.
Frequently asked: what is a full-mark explanation?
A strong explanation states the starting point, shows north, measures clockwise, gives a properly written three-figure direction, and includes a distance with appropriate units if requested. For a right-triangle calculation, state why sine, cosine or tangent is used and whether its angle is already the clockwise bearing or requires quadrant adjustment.
Bukit Timah Mathematics learning and official references
The unchanged eduKateSG Mathematics teaching benchmark describes three-pupil tutorials, approximately 1.5 hours weekly, and guided corrections at 8 Fourth Avenue, Singapore 268674 near Sixth Avenue MRT. Bukit Timah identifies families who may travel to that established venue; it does not imply another location or an available booking slot.
G3 is a Mathematics subject level within Full Subject-Based Banding. The SEAB SEC overview and 2027 K310 official Mathematics syllabus describe longer-term progression. Bearings and trigonometric extensions may be taught later than Secondary 2, depending on the school. Follow the student’s current assessment scope.
For related methods, see map scales and distances, Pythagoras and right triangles, and SOH CAH TOA trigonometry. They cover skills that this bearings guide applies to direction.
To discuss a suitable small-group tutorial, contact eduKate Singapore or enquire on WhatsApp. Bring one recently marked geometry question and the next assessed chapters. Confirm current class suitability, times and vacancies.
Editorial review: 26 October 2026. Every map, position and journey is a hypothetical classroom diagram, not a real location, navigational route or live travel estimate. Numerical trigonometric results rounded to the nearest degree are indicated as such.