NSWMaths Standard 2Syllabus dot point

How are bearings and radial surveys used to find distances and directions in navigation and surveying?

Use compass and true bearings, and radial surveys, to solve practical navigation and surveying problems

A focused answer to the HSC Maths Standard 2 dot point on bearings and radial surveys. Compass vs true bearings, back-bearings, the structure of a radial survey, and worked Australian navigation examples using the sine and cosine rules.

Generated by Claude OpusReviewed by Better Tuition Academy8 min answerUpdated 2026-05-20

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What this dot point is asking

NESA wants you to read and write both compass and true bearings, draw radial-survey diagrams, and combine bearings with the sine rule, cosine rule and Pythagoras to find distances and directions in navigation and surveying problems.

The answer

True bearings

A true bearing is measured clockwise from north, in three digits, from $000\degree$ to $360\degree$ .

North: $000\degree$
East: $090\degree$
South: $180\degree$
West: IMATH_8

So a bearing of $075\degree$ means $75\degree$ clockwise from north. Always write three digits ( $075\degree$ , not $75\degree$ ).

Compass bearings

A compass bearing uses a primary direction (N or S) followed by an angle, then E or W. For example, N $30\degree$ E means $30\degree$ east of due north.

Compass bearings are less common in HSC than true bearings, but you should be able to convert between them.

Back-bearings

If the bearing of $B$ from $A$ is $\theta$ , then the bearing of $A$ from $B$ is $\theta + 180\degree$ (subtract $360\degree$ if the result exceeds $360\degree$ ).

So if $B$ is at bearing $075\degree$ from $A$ , then $A$ is at bearing $255\degree$ from $B$ .

Radial surveys

A radial survey records the distances and true bearings of several points from a single central station. The data is usually given as a table or a diagram.

To find distances or angles between two of the surveyed points (not involving the centre), you usually need the cosine rule.

To find the interior angle of the triangle at the central station, take the difference between the two bearings.

Common geometry pitfalls

The interior angle of a triangle at the bend of a path is not the change of bearing. It is the supplement of the change of bearing.

Worked logic: you arrive at $A$ heading on bearing $075\degree$ . You leave heading on bearing $145\degree$ . The change in heading is $70\degree$ . The original direction continued (the "straight ahead") and the new direction make a $70\degree$ angle, so the interior angle at $A$ in the triangle $PAB$ is $180\degree - 70\degree = 110\degree$ .

If the change is small (you barely turn), the interior angle is large (close to $180\degree$ ). If the change is large (you turn nearly around), the interior angle is small.

Worked example

Two-leg journey

A walker leaves base camp and walks $5$ km on bearing $060\degree$ to point $A$ , then $7$ km on bearing $150\degree$ to point $B$ . Find the straight-line distance from base camp to $B$ .

Interior angle at $A$ : turn from $060\degree$ to $150\degree$ is $90\degree$ , so interior angle at $A$ is $180 - 90 = 90\degree$ .

(In this case the interior angle happens to be $90\degree$ , so Pythagoras applies.)

$\text{dist}^2 = 5^2 + 7^2 = 25 + 49 = 74$ .

$\text{dist} = \sqrt{74} \approx 8.6$ km.

Coastal survey (Australian context)

From the lighthouse at Norah Head on the NSW Central Coast, two ships are observed: ship $X$ at bearing $060\degree$ true, distance $4$ km, and ship $Y$ at bearing $135\degree$ true, distance $6$ km.

Distance $XY$ : in triangle (lighthouse) $XY$ , the interior angle at the lighthouse is $135 - 60 = 75\degree$ .

$XY^2 = 4^2 + 6^2 - 2 \times 4 \times 6 \cos 75\degree = 16 + 36 - 48 \times 0.2588 = 52 - 12.42 = 39.58$ .

$XY \approx \sqrt{39.58} \approx 6.29$ km.

Back-bearing

If a yacht observes the headland at bearing $250\degree$ , what bearing is the yacht from the headland?

Back-bearing: $250 - 180 = 070\degree$ .

(Subtraction works when the original bearing is $180\degree$ or more; add $180$ otherwise.)

Past exam questions, worked

Real questions from past NESA papers on this dot point, with our answer explainer.

2022 HSC Q275 marksA ship leaves port

P

and sails

40

km on a bearing of

075\degree

to point

A

, then

55

km on a bearing of

145\degree

to point

B

. Find the distance and bearing of

B

from

P

Show worked answer →

The interior angle at $A$ in triangle $PAB$ is the supplement of the change in direction. From $075\degree$ to $145\degree$ is a turn of $70\degree$ , so the interior angle at $A$ is $180\degree - 70\degree = 110\degree$ .

Distance $PB$ by the cosine rule:

$PB^2 = 40^2 + 55^2 - 2 \times 40 \times 55 \times \cos 110\degree = 1600 + 3025 - 4400 \times (-0.3420) = 4625 + 1505 = 6130$ .

$PB \approx \sqrt{6130} \approx 78.3$ km.

For the bearing, find angle $APB$ by the sine rule: $\sin(\angle APB) = \frac{55 \sin 110\degree}{78.3} \approx \frac{55 \times 0.9397}{78.3} \approx 0.660$ , so $\angle APB \approx 41.3\degree$ .

Bearing of $B$ from $P$ : $075\degree + 41.3\degree = 116.3\degree$ , round to $116\degree$ .

Markers reward the interior angle at $A$ (with the supplement step shown), the cosine rule, the sine rule for the angle at $P$ , and the final bearing.

2021 HSC Q264 marksFrom point

A

, point

B

is at a bearing of

030\degree

and a distance of

200

m. From point

A

, point

C

is at a bearing of

120\degree

and a distance of

150

m. Find the distance

BC

Show worked answer →

In triangle $ABC$ , the angle at $A$ is the difference in bearings: $120\degree - 030\degree = 90\degree$ .

Use Pythagoras (or the cosine rule with $\cos 90\degree = 0$ ):

$BC^2 = 200^2 + 150^2 = 40000 + 22500 = 62500$ .

$BC = \sqrt{62500} = 250$ m.

Markers reward the angle at $A$ identified as the difference of bearings, recognising it as a right angle, and the distance. A radial-survey diagram is expected.

What this dot point is asking

The answer

True bearings

Compass bearings

Back-bearings

Radial surveys

Common geometry pitfalls

Past exam questions, worked

Related dot points