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, the angle is written with three digits, so east is $090^\circ$, south is $180^\circ$, and west is $270^\circ$.

That single rule — start at north, turn clockwise, stop at the line of travel — drives almost every bearing calculation.

Why bearings are measured this way

The whole-circle system exists so a direction can be stated unambiguously with one number. Measuring clockwise from north gives a fixed reference everyone shares:

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

So a bearing of $045^\circ$ means $45^\circ$ clockwise from north, and a bearing of $210^\circ$ means turning clockwise from north until the angle reaches $210^\circ$, which lands past south. Because the reference and direction of turn are fixed, two people reading the same bearing always face the same way.

Worked example: a reverse bearing

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 straight line in the opposite direction.

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

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

The rule: if the original bearing is less than $180^\circ$, add $180^\circ$; if it is $180^\circ$ or more, subtract $180^\circ$.

It also helps to translate common bearings into plain language: $030^\circ$ is $30^\circ$ east of north, $120^\circ$ is $30^\circ$ south of east, and $300^\circ$ is $60^\circ$ west of north. That is a picture aid only; the official bearing stays the three-figure clockwise angle from north.

Try it yourself, then check the answer

Point $C$ is on a bearing of $140^\circ$ from point $D$. Find the bearing of $D$ from $C$. Since $140^\circ < 180^\circ$, add $180^\circ$ to get $320^\circ$. Sketch both points, draw a north line at each, and confirm your answer points back along the same straight line.

Calculation traps to watch for

• Measuring from east instead of north. Standard bearings start from north, not the horizontal axis.
• Turning the wrong way. Three-figure bearings are clockwise; measuring anticlockwise gives the wrong direction.
• Dropping the three-digit format. Write $040^\circ$, not $40^\circ$; the value is the same angle, but the three-digit form is part of the notation.
• Using the wrong starting point. The bearing of $B$ from $A$ is measured at $A$, not at $B$, and the reverse changes the bearing by $180^\circ$.

Where bearings are used

Bearings appear in map reading, marine navigation, aviation, surveying, and geometry problems about direction. They are especially useful when a direction must be communicated precisely with one angle instead of a vague phrase like "roughly north-east." In school math they often combine with triangles and trigonometry: once you know a direction and one or two distances, you can find unknown sides or angles.