Compass & True Bearings
Read and write true and compass bearings for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-04). A bearing gives a direction as an angle: a true bearing such as \(125^\circ\text{T}\) is measured clockwise from north, while a compass bearing such as \(\text{S}55^\circ\text{E}\) starts at north or south and turns toward east or west.
You will read bearings off a diagram, write true bearings with three figures, convert from one type to the other, and find a back bearing — the skills bushwalkers, sailors and pilots use to say where they are heading.
Theory
A true bearing is measured clockwise from north and written with three figures, such as \(125^\circ\text{T}\). A compass bearing starts at north or south and turns toward east or west, such as \(\text{S}55^\circ\text{E}\). This Year 12 Standard 1 (NSW) guide shows how to convert between them and find back bearings.
A bearing gives a direction as an angle. A true bearing is the angle turned clockwise from north, written with three figures and a T, for example \(045^\circ\text{T}\), \(160^\circ\text{T}\) or \(305^\circ\text{T}\). The key directions are N \(=000^\circ\text{T}\), E \(=090^\circ\text{T}\), S \(=180^\circ\text{T}\) and W \(=270^\circ\text{T}\).
A compass bearing starts at N or S, then turns the stated angle toward E or W, for example \(\text{N}45^\circ\text{E}\), \(\text{S}20^\circ\text{E}\) or \(\text{N}55^\circ\text{W}\). To convert, find the quadrant first.
The back (reverse) bearing is the direction back again. It differs by \(180^\circ\): add \(180^\circ\) if the bearing is under \(180^\circ\), and subtract \(180^\circ\) if it is over.
Find the quadrant first, then convert:
| Quadrant | Compass to true | True to compass |
|---|---|---|
| NE | \(\text{N}a^\circ\text{E} = a^\circ\text{T}\) | \(\text{N}\,(\text{bearing})\,\text{E}\) |
| SE | \(\text{S}a^\circ\text{E} = (180-a)^\circ\text{T}\) | \(\text{S}\,(180-\text{bearing})\,\text{E}\) |
| SW | \(\text{S}a^\circ\text{W} = (180+a)^\circ\text{T}\) | \(\text{S}\,(\text{bearing}-180)\,\text{W}\) |
| NW | \(\text{N}a^\circ\text{W} = (360-a)^\circ\text{T}\) | \(\text{N}\,(360-\text{bearing})\,\text{W}\) |
The back bearing differs by \(180^\circ\):
How to convert a bearing
- Sketch a compass cross (N, E, S, W) at the starting point.
- Draw the direction and decide which quadrant it is in.
- Measure the angle the way the answer needs: clockwise from north for a true bearing, from N or S toward E or W for a compass bearing.
- Write it correctly: three figures and T for a true bearing, or letter-angle-letter for a compass bearing.
Sketch the compass. \(\text{S}40^\circ\text{W}\) is in the SW quadrant: start at due south and turn \(40^\circ\) further clockwise.
| \(\text{Due south}\) | \(=\) | \(180^\circ\) |
| \(180^\circ+40^\circ\) | \(=\) | \(220^\circ\) |
The true bearing is \(220^\circ\text{T}\).
Sketch the compass. \(290^\circ\) is between W \((270^\circ)\) and N \((360^\circ)\), so it is in the NW quadrant. Measure from north.
| \(360^\circ-290^\circ\) | \(=\) | \(70^\circ \text{ west of north}\) |
The compass bearing is \(\text{N}70^\circ\text{W}\).
Sketch both points with a north line at each. The way back differs by \(180^\circ\). As \(118^\circ < 180^\circ\), add.
| \(118^\circ+180^\circ\) | \(=\) | \(298^\circ\) |
The bearing of \(B\) from \(J\) is \(298^\circ\text{T}\).
Sketch both points. Change to a true bearing, reverse it, then name the quadrant.
| \(\text{N}55^\circ\text{E}\) | \(=\) | \(055^\circ\text{T}\) |
| \(055^\circ+180^\circ\) | \(=\) | \(235^\circ\text{T}\) |
| \(235^\circ-180^\circ\) | \(=\) | \(55^\circ\text{ west of south}\) |
(i) \(235^\circ\text{T}\) (ii) \(\text{S}55^\circ\text{W}\)
Common pitfalls
Frequently asked questions
What is the difference between a true bearing and a compass bearing?
A true bearing is measured clockwise from north and written with three figures, such as 125 degrees T. A compass bearing starts at north or south and turns toward east or west, such as S55 degrees E. They can describe the same direction.
How do you convert a compass bearing to a true bearing?
Find the quadrant. In the NE quadrant the true bearing is the angle itself. In the SE quadrant subtract the angle from 180. In the SW quadrant add the angle to 180. In the NW quadrant subtract the angle from 360.
Why are true bearings written with three figures?
Writing every true bearing with three digits, such as 075 degrees T instead of 75 degrees T, avoids confusion when bearings are read aloud or written quickly. It is the standard way to write them.
How do you find a back bearing?
The back bearing is the direction back to the start. Add 180 degrees if the bearing is less than 180 degrees, or subtract 180 degrees if it is more than 180 degrees.
Can a compass bearing start from east or west?
No. A compass bearing always starts from north or south and then turns toward east or west, such as N70 degrees W or S40 degrees W.