How to Learn Bearings and Scale Drawings Quickly and Correctly | Quick Notes

To learn Bearings and Scale Drawings quickly and correctly, remember two simple ideas: a bearing tells you the direction to travel, while a scale tells you how a drawing’s length compares with real distance. Bearings are measured clockwise from north and usually written using three figures, such as 045° or 270°. Scale drawings use a consistent ratio so you can measure real distances using a map or plan. These eduKateSG Mathematics Quick Notes show how to read directions, calculate reverse bearings and convert map lengths without guessing.

• Start at north: a bearing is measured clockwise from the north line at the starting point.
• Write three figures: use 035°, 090°, 180° or 315° rather than ambiguous shorthand.
• Reverse direction: for opposite bearings, add or subtract 180° and keep within 0° to less than 360°.
• Read the scale: 1 : 50,000 means 1 unit on the map represents 50,000 of the same units in reality.
• Convert units: change centimetres to metres or kilometres after calculating.
• Check the diagram: make sure north is vertical at the chosen point and the measured distance matches the required route.

First, learn bearings by standing at the starting point and imagining a compass. Straight north has bearing 000°, east is 090°, south is 180° and west is 270°. North-east lies halfway between north and east, so its bearing is 045°. North-west is 315°. The direction must be measured clockwise from north, not from the east or from whichever line is easiest to see. When given two locations A and B, “bearing of B from A” means put the compass at A and point towards B.

Second, work with reverse bearings as opposite directions. If the bearing of B from A is 065°, then the bearing of A from B is 065° + 180° = 245°. If the first bearing is 230°, subtract 180° to find its reverse: 050°. This rule assumes you are working in an ordinary planar bearing problem where the north lines at the two points are parallel. A clear sketch with north marked at both points helps you see why an opposite direction is a half-turn away.

Third, read a scale ratio using identical units. On a map marked 1 : 50,000, every 1 cm drawn represents 50,000 cm in reality. Convert 50,000 cm to 500 m, or 0.5 km. Thus a straight map segment of 6 cm represents 3 km. It is tempting to multiply 6 by 50,000 and call the result kilometres, but the product is first in centimetres. Always write the units at the conversion step, because scale questions often test unit control more than difficult arithmetic.

Worked example: find a bearing from a diagram

You are at point A and point B lies east of A. The bearing of B from A is 090°. If C lies 30° east of north from A, the clockwise bearing of C from A is 030°. If D lies 20° north of west from A, west is 270° and moving 20° toward north increases the clockwise compass angle, giving 290°. Check each answer by describing the compass direction aloud. Writing three figures helps avoid confusing a 30° bearing with 300°.

Worked example: find a return bearing

A boat travels from harbour H to buoy B on a bearing of 140°. What is the bearing of H from B? The return journey points exactly the opposite way, so add 180°: 140° + 180° = 320°. The return bearing is therefore 320°. This is not automatically the bearing of another route between the same two named places if the return path bends; the rule describes the straight direction between the two points.

Worked example: scale 1 : 25,000

A trail measures 4 cm as a straight segment on a map with scale 1 : 25,000. Actual straight-line distance = 4 × 25,000 = 100,000 cm. Since 100 cm = 1 m, this is 1,000 m = 1 km. To check, 1 cm at this scale represents 250 m, so 4 cm represents 4 × 250 m = 1,000 m. If the trail curves, a straight-line map segment is not necessarily the length of the walking route; measure the path that the question actually requests.

Worked example: find a drawing length from real distance

A school building is 60 m long and will be shown on a plan at scale 1 : 500. Convert 60 m to centimetres: 60 × 100 = 6,000 cm. Divide by 500 to get the drawing length: 12 cm. Check by reversing: 12 cm on the plan represents 12 × 500 = 6,000 cm = 60 m. The direction of the conversion matters: map-to-real normally multiplies by the scale denominator, while real-to-map divides after matching units.

How to draw a three-figure bearing correctly

1. Mark the starting point and draw a north arrow through it.
2. Place the protractor’s centre at the starting point and align zero with north.
3. Measure the stated angle clockwise.
4. Draw a ray through the measured angle and label the target direction.
5. Write the bearing with three digits and a degree symbol.
6. If a distance is supplied, measure that distance along the ray using the given drawing scale.

A combined bearing and scale problem

A point B is 6 km from point A on a bearing of 060°. You must draw the journey using a map scale of 1 : 100,000. At this scale, 1 cm represents 100,000 cm = 1 km. Draw the north line at A, measure 060° clockwise and draw a ray 6 cm long. Label the endpoint B. The reverse bearing of A from B is 240°. Both the drawn length and the compass direction must be correct for the scale drawing to represent the journey.

Why scale factors affect areas differently

If a drawing’s linear dimensions are scaled by a factor of k, its areas are scaled by k². For example, a square drawn 2 cm by 2 cm at scale 1 : 100 represents a real square 2 m by 2 m. The map area is 4 cm², while the real area is 4 m² or 40,000 cm². The area ratio is therefore 1 : 10,000, not 1 : 100. Most basic map questions concern distances, but recognising the different area scale prevents a serious mistake in floor plans and models.

Common mistakes and quick repairs

• Measuring anti-clockwise: bearings start at north and go clockwise.
• Writing 45° instead of 045°: use the standard three-figure form.
• Finding the bearing from the wrong point: “B from A” means stand at A.
• Using 180° − the bearing for a return journey: a reverse direction changes by 180°, not by a generic supplementary angle.
• Mixing metres and centimetres in a scale ratio: use identical units before multiplying or dividing.
• Measuring a curved route as a straight segment: follow the specific path described.
• Ignoring the north arrow: check whether the drawing itself has been rotated.

Mini-test: answer without guessing

1. What is the three-figure bearing for south-east?
2. The bearing of B from A is 075°. Find the reverse bearing.
3. On a 1 : 50,000 map, a straight-line distance is 8 cm. Find the actual distance in kilometres.
4. At scale 1 : 200, how long on the plan is a wall that is 10 m long?
5. The bearing of a point P from O is 280°. Is P north or south of the west direction?

Answers and explanations

1. South-east is halfway from east (090°) to south (180°): 135°. 2. 075° + 180° = 255°. 3. 8 × 50,000 = 400,000 cm = 4 km. 4. 10 m = 1,000 cm; 1,000 ÷ 200 = 5 cm. 5. P lies north of west because 280° is 10° clockwise beyond due west (270°), towards north.

Your ten-minute bearings and scale routine

1. Minutes 1–2: draw north, east, south and west and label the three-figure bearings.
2. Minutes 3–4: sketch two bearings and find their reverse directions.
3. Minutes 5–7: convert a measured map distance to metres or kilometres.
4. Minute 8: calculate a drawing distance from a real measurement.
5. Minutes 9–10: combine one bearing with one scale and check your protractor direction and units.

The easiest way to stay accurate is to separate the tasks: first get the direction right with north and a clockwise angle, then get the distance right with a consistent scale. Each has its own check.

Continue learning with eduKateSG Quick Notes

Use these short connected lessons: Angles · Trigonometric Ratios · Similarity and Congruence · Vectors. Solve one changed question tomorrow without looking at the worked solution.

For the longer teaching pathway, see Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials. The original reference remains unchanged.