Bearings and Scale Drawing

Lesson 4 of 4: Geometric Construction

In this lesson:

• Read and write three-figure bearings measured clockwise from North
• Measure and draw bearings with a protractor
• Make and interpret scale drawings
• Combine bearings and distances to solve navigation problems

What You Will Be Able to Do

1. Read and write three-figure bearings — clockwise from North
2. Measure a bearing off a diagram; draw one with a protractor
3. Make and interpret scale drawings — convert lengths to real distances
4. Solve problems: bearing + distance to scale, including the return bearing

Which Direction Is Exact Enough?

A pilot radios the tower: "I'm heading north-east."

The controller needs the exact direction to track the plane.

• North-east is somewhere between N and E
• But how far between? Halfway? Three-quarters of the way?
• "North-east" is not precise enough for navigation

What would make a direction unambiguous?

You Already Know the Compass

Where would you place NE? How many degrees from North?

Introducing the Three-Figure Bearing System

A bearing is an exact direction: always clockwise from North, always written with three figures.

• N = 000°    E = 090°    S = 180°    W = 270°
• Write 045°, not 45°    Write 007°, not 7°

Quick Check: Cardinal and Half-Cardinal Bearings

What is the bearing of SE from a point?

What is the bearing of NW from the same point?

Think before advancing — write the three-figure answer.

Why the North Line Comes First

A bearing is an angle — every angle needs a starting line.

• A North line can be drawn at any point on any map
• It is the same reference everywhere in the world
• Every bearing diagram must start: draw North first

North first — always.

How to Draw a Bearing: 120°

Step 1: Mark point . Draw a North line (vertical arrow, upward)

Step 2: Place the protractor centre on , 0° on the North line

Step 3: Count clockwise to 120° and make a mark

Step 4: Draw line from through the mark

Spot the Errors in This Bearing

A student drew this bearing diagram:

• They placed the protractor with 0° on the East line
• They measured the angle anticlockwise

Name both errors. What should they have done instead?

Measuring a Bearing from a Diagram

To measure the bearing of from :

1. Draw a North line at point
2. Place protractor centre on , 0° on the North line
3. Read clockwise to the line

The dial reads 70° — write it as 070°. Three figures, always.

Practice: Draw a Bearing Yourself

Draw a bearing of 210° from point .

Use a protractor and ruler. Write your three-figure answer on the diagram.

Check: 210° is past South (180°) going toward West — does your diagram show that?

Bearing Alone Does Not Fix a Position

Direction alone does not fix a position on a map.

• Bearing 050° from town — destination could be 2 km or 200 km away
• To fix a point: bearing and distance
• Distance on a drawing ≠ distance on the ground — you need a scale

What Is a Map Scale?

• A scale links drawing length to real distance
• 1 cm : 2 km → 1 cm on paper = 2 km on the ground
• Ratio form: 1 : 200 000 (matched units, same idea)
• Keep units on every line — never mix km and cm

Converting Distances with the Scale

Scale: 1 cm : 2 km

Real → Drawing (divide):

Drawing → Real (multiply):

Why Divide Real → Drawing?

The drawing is smaller than reality — you are shrinking.

• Shrinking means the paper length is a smaller number than the real distance
• Divide gives a smaller answer — that is the right direction

Sanity check: drawn length should always be smaller than the real distance in km.

Scale Practice: Real to Drawing and Back

Scale: 1 cm : 1 km

A road is 5 km long. How long is it on the drawing?

A drawn line is 3 cm. How far is that in real life?

Work out both before advancing.

Bearing + Scale = Fixed Position

Bearing gives direction. Scale gives distance.

Together, they fix any point on a map:

• Draw the bearing from the start point
• Measure the scaled distance along that bearing line
• Mark the destination

This is the payoff: navigation on paper.

Worked Example: Locating Point B

Problem: is 6 km from on a bearing of 050°. Scale: 1 cm : 1 km.

Back bearing of from : 050° + 180° = 230°

The Back Bearing Rule: ±180°

The back bearing differs by 180°:

• Under 180°: add 180°    → 050° + 180° = 230°
• 180° or more: subtract 180°    → 200° − 180° = 020°
• Answer must be in 000°–360°

Return = opposite direction = half a full turn.

Your Turn: Full Navigation Problem

Problem: Town is 8 km from point on a bearing of 135°. Scale: 1 cm : 2 km.

1. Draw a scale drawing and mark town
2. State the bearing of from

No hints — work the full procedure from blank paper.

Five Common Bearing and Scale Errors

Wrong axis — measure from North, not East or horizontal

Anticlockwise — always clockwise from North

Dropped leading zero — write 070°, not 70°

Multiplied real → drawing — divide, do not multiply

Same bearing back — return bearing differs by 180°

What You Can Now Do

✓ Bearing — exact direction, clockwise from North, three figures

✓ Scale — divide real → drawing; multiply drawing → real

✓ Bearing + distance fixes any point on a map

✓ Back bearing differs by 180° — return is always the opposite direction

Topic 8: Geometric Construction Complete

Geometric Construction — Topic 8 — finished.

You've used protractor and ruler to:

• Measure and construct angles (Lesson 1)
• Construct accurate geometric figures (Lessons 2–3)
• Fix locations using bearings and scale (Today)

These instruments, handled accurately, are enough to solve real navigation problems.