Scale Drawing & Navigation

Video coming soon2 worked examples

Theory

The Golden Rule: Bearings must always be measured clockwise from North, and they must always be written using exactly three digits (e.g., 045°, not 45°). When plotting a multi-stage journey, you must draw a completely new, perfectly vertical North line at every single checkpoint before measuring your next angle.

1. Using a Scale

A scale drawing is a proportional reduction of a real-life space.

Real-life to the Page: To find out how long a line should be on your page, convert the real-life distance to match the scale's units, then divide by the scale factor. (Example: For a scale of 1 cm:10 km, a 60 km flight is drawn as $60 \div 10 = 6 \text{ cm}$ ).
Page to Real-life: To find a real-life distance, measure the line on the page in centimetres and multiply by the scale factor.

2. Constructing the Diagram

You must use a ruler to draw lines accurately to the nearest millimetre (±2 mm tolerance) and a protractor to draw angles accurately to the nearest degree ( $\pm 2^\circ$ tolerance).

Start at the given point.
Align the centre of your protractor with the start point, ensuring the $0^\circ$ line points straight up along the North line.
Measure the angle clockwise and mark it.
Draw the scaled length through that mark using your ruler.

3. The Return Journey

You will frequently be asked to find the bearing of the return journey (e.g., heading back to the start from the final location).

You must draw a new North line at your final location.
Place your protractor on this new North line and measure clockwise all the way around until you hit the line that leads back to the start.
(Tip: If the return line points towards the bottom-left, the bearing will be greater than $180^\circ$ , so you may need to measure the angle past $180^\circ$ and add it on).

⚠️ Common Examiner Traps

The "Return Journey Bearing" Trap: a common mistake on the return journey is not measuring its bearing at all, or measuring it but not stating the angle as a three-figure bearing. You must ensure your final answer has three digits!
The "Incomplete Diagram" Trap: a common mistake is to calculate the scaled lengths in centimetres correctly but then fail to construct a diagram of the entire course. You must follow through and actually draw every single leg of the journey with a ruler and protractor to secure the marks.
The "Missing North Line" Trap: Candidates frequently try to measure the second bearing using the angle of the previous line they just drew. You must draw a completely new, straight-up North line at the end of leg 1 before measuring the angle for leg 2.

Worked examples

Example 1

The start of an orienteering course is being planned.

Competitors leave the start point and run on a bearing of $335^\circ$ for 400 metres to checkpoint A.
From checkpoint A they then run on a bearing of $030^\circ$ for 320 metres to checkpoint B.

(a) Construct a scale drawing to illustrate this part of the course. Use a scale of 1 cm:100 m. (3 marks)

(b) Use your scale drawing to determine the distance and bearing of the start point from checkpoint B. (2 marks)

Step 1: Calculate the lengths to be drawn on the page by dividing the real-life distances by the scale factor (100).

Leg 1: $400 \text{ m} \div 100 = 4 \text{ cm}$ .
Leg 2: $320 \text{ m} \div 100 = 3.2 \text{ cm}$ .

Step 2 (Drawing Leg 1): Place the protractor on the start point's North line. Because $335^\circ$ is reflex, measure $25^\circ$ anti-clockwise (since $360 - 335 = 25$ ). Draw a line exactly 4 cm long. Label the end "Checkpoint A".

Step 3 (Drawing Leg 2): Draw a new vertical North line at Checkpoint A. Measure $30^\circ$ clockwise from this new line. Draw a line exactly 3.2 cm long. Label the end "Checkpoint B".

Step 4 (Return Distance): Draw a straight dotted line from Checkpoint B back to the Start. Measure it with a ruler. (Let's assume it measures 5.4 cm). Multiply by the scale factor to find the real-life distance.

5.4 \text{ cm} \times 100 = 540 \text{ metres}

Step 5 (Return Bearing): Draw a final North line at Checkpoint B. Measure the angle clockwise from North all the way around to the dotted return line. (Let's assume it points downwards/left, measuring $201^\circ$ ).

Final Answer (b): The distance is 540 m on a bearing of $201^\circ$ .

Example 2

A surveyor produces a sketch of a radio mast in a field.

The horizontal distance from the surveyor to the base of the mast is 490 metres.
The angle of elevation from the ground to the top of the mast is $20^\circ$ .
The mast and the ground form a right-angled triangle.

Using a scale of 1 cm:20 m, make a scale drawing of the triangle and use it to find the real-life height of the radio mast. (4 marks)

Step 1: Calculate the horizontal length for the drawing by dividing by the scale factor.

490 \text{ m} \div 20 = 24.5 \text{ cm}

Step 2 (Drawing): Draw a horizontal line exactly 24.5 cm long with a ruler. At the right-hand end, use a protractor to draw a $20^\circ$ angle pointing upwards and leftwards.

Step 3 (Drawing): At the left-hand end, draw a perfect $90^\circ$ vertical line going straight up to represent the mast. Extend the $20^\circ$ slanted line until it crosses the vertical line to complete the triangle.

Step 4 (Calculate Height): Measure the vertical line you drew. (It should measure exactly 8.9 cm). Multiply this page length by the scale factor to find the real-life height.

8.9 \text{ cm} \times 20 = 178 \text{ metres}

Final Answer: The real-life height of the radio mast is 178 metres.

Ready to practise?

9 Scale Drawing and Navigation questions with answers and video solutions