Bearings — Three-Figure Bearings & Navigation

A bearing is a direction written as an angle measured clockwise from north. In the three-figure system used in school math and basic navigation, you write the angle with three digits: east is $090^\circ$, south is $180^\circ$, and west is $270^\circ$.

That is the core idea behind most bearing questions. Start at north, turn clockwise, and stop when you reach the line of travel or the line joining the two points.

What three-figure bearings mean

In the usual school and navigation setting, bearings use the whole-circle system:

$$0^\circ \text{ to } 360^\circ$$

measured clockwise from north.

So:

• $000^\circ$ or $360^\circ$ points north
• $090^\circ$ points east
• $180^\circ$ points south
• $270^\circ$ points west

For example, a bearing of $045^\circ$ means $45^\circ$ clockwise from north. A bearing of $210^\circ$ means the direction is past south, turning clockwise from north until the angle reaches $210^\circ$.

How to measure or draw a bearing

Use this order every time:

1. Mark the starting point.
2. Draw or identify the north line at that point.
3. Measure clockwise from north.
4. Write the angle with three digits.

The starting point matters. A bearing of $B$ from $A$ is measured at $A$, not at $B$.

Worked example: reverse bearing

Suppose a boat travels from harbor $A$ to buoy $B$ on a bearing of $065^\circ$. Find the bearing of $A$ from $B$.

This is the reverse bearing. In the whole-circle system, the reverse direction is always $180^\circ$ away because it points along the same line in the opposite direction.

Since $065^\circ < 180^\circ$, add $180^\circ$:

$$065^\circ + 180^\circ = 245^\circ$$

So the bearing of $A$ from $B$ is

$$245^\circ$$

If the original bearing is less than $180^\circ$, add $180^\circ$. If it is $180^\circ$ or more, subtract $180^\circ$ instead.

A quick way to picture common bearings

Some bearings are easier to understand if you translate them into plain language:

• $030^\circ$ means $30^\circ$ east of north
• $120^\circ$ means $30^\circ$ south of east
• $300^\circ$ means $60^\circ$ west of north

This is only a picture aid. The official bearing is still written as the three-figure clockwise angle from north.

Common mistakes with bearings

Measuring from east instead of north

Bearings do not start from the horizontal axis unless the problem says otherwise. In standard bearing questions, you start from north.

Turning the wrong way

Three-figure bearings are measured clockwise. If you measure anticlockwise, you get the wrong direction.

Forgetting the three-digit format

Write $040^\circ$, not $40^\circ$. The value is the same angle, but the three-digit format is part of the notation.

Using the wrong starting point

The bearing of $B$ from $A$ is not the same as the bearing of $A$ from $B$. Reverse the direction and the bearing changes by $180^\circ$ in the whole-circle system.

When bearings are used

Bearings appear in map reading, marine navigation, aviation, surveying, and geometry problems about direction. They are especially useful when a direction needs to be communicated clearly with one angle instead of a vague phrase like "roughly north-east."

In school math, bearings often combine with triangles and trigonometry. Once you know the direction and one or two distances, you can start finding unknown sides or angles.

Try a similar problem

Point $C$ is on a bearing of $140^\circ$ from point $D$. Find the bearing of $D$ from $C$.

Sketch both points, draw a north line at each one, and check that your answer points back along the same straight line. That quick sketch catches most bearing mistakes.