Bearings are a precise way to describe direction using angles measured from North. They are used in navigation, surveying, maps, aviation, and geometry problems involving position. Unlike informal compass directions such as northeast, a bearing gives a numerical direction that can be measured and calculated.
This makes it easier to communicate exact routes and solve diagrams with angles and distances.
A bearing is measured clockwise from North and is usually written as a three-digit angle from 000° to 360°. For example, East is 090°, South is 180°, and West is 270°. Standard position angles in coordinate geometry are usually measured counterclockwise from the positive x-axis, so converting between the two systems requires careful attention to the starting direction and rotation direction.
In triangle and map problems, bearings often combine with scale drawings, angle facts, and trigonometry.
Understanding Geometry: Bearings and Compass Directions
A bearing belongs to a direction from one place to another. This matters when a journey has several legs. Suppose a walker travels from A to B, then from B to C.
The direction from A to B is drawn from A, while the direction from B to C is drawn from B. Students sometimes keep measuring every new direction from the first point. That creates a diagram that looks neat but represents a different route.
Marking a small north arrow at each important point helps. On a flat map, all these north lines are parallel, so they give a reliable reference for each measurement.
The direction back along the same straight path needs special care. A path has two possible travel directions, separated by half a turn. If one person observes a lighthouse in a particular direction, the lighthouse observer sees that person in the opposite direction.
This idea is useful in problems where two places see each other, or when a vehicle must return on the same route. It can reveal mistakes quickly.
A proposed return direction that is close to the original direction, rather than roughly half a turn away, is almost certainly wrong. In a triangle diagram, this opposite direction is often needed before ordinary interior angle facts can be used.
Bearings connect naturally to coordinates and trigonometry. Imagine north as the vertical direction on a map and east as the horizontal direction. A journey can be split into a northward part and an eastward part.
For a distance d at bearing b, the eastward displacement is d times the sine of b. The northward displacement is d times the cosine of b. These relationships may seem reversed at first because the bearing begins from north, not from the horizontal axis used in many coordinate graphs.
The signs matter too. A direction toward the west gives a negative eastward displacement.
A direction toward the south gives a negative northward displacement. Adding the separate displacements for every leg gives the final position.
Real navigation adds a practical complication. A paper map may use true north, while a compass needle points toward magnetic north. The gap between them is called magnetic declination and changes with location and time.
Small school problems usually state which north line to use, or they treat them as the same. In surveying, aviation, hiking, and sea travel, mixing them can put a route noticeably off course.
Students should read labels carefully, check the scale before turning a measured length into a real distance, and use a protractor from its correct centre. A line drawn one or two degrees inaccurately can create a large position error after a long distance.
Key Facts
- Bearings are measured clockwise from North.
- A three-figure bearing always uses three digits, such as 045°, 120°, or 305°.
- North = 000° or 360°, East = 090°, South = 180°, West = 270°.
- For a standard angle θ measured counterclockwise from East, bearing B = (90° - θ) mod 360°.
- For a bearing B, the standard angle θ = (90° - B) mod 360°.
- Opposite or back bearing = B + 180° if B < 180°, and B - 180° if B ≥ 180°.
Vocabulary
- Bearing
- A bearing is an angle that gives direction, measured clockwise from North.
- Compass rose
- A compass rose is a diagram that shows the main directions North, East, South, and West, often with intermediate directions.
- Three-figure bearing
- A three-figure bearing is a bearing written with three digits, such as 007° or 250°.
- Back bearing
- A back bearing is the direction exactly opposite a given bearing, found by adding or subtracting 180°.
- Standard angle
- A standard angle is usually measured counterclockwise from the positive x-axis in coordinate geometry.
Common Mistakes to Avoid
- Measuring bearings counterclockwise is wrong because bearings are always measured clockwise from North, not from East or from the positive x-axis.
- Writing 45° instead of 045° is incorrect for a three-figure bearing because bearings below 100° should include leading zeros.
- Confusing 000° and 180° is wrong because 000° points North while 180° points South, which are opposite directions.
- Finding a back bearing by always adding 180° is incomplete because bearings must stay between 000° and 360°, so subtract 180° when the original bearing is 180° or greater.
Practice Questions
- 1 A boat travels on a bearing of 135°. Which main compass directions is this between, and how many degrees south of East is it?
- 2 Convert the standard angle 210° into a bearing, assuming the standard angle is measured counterclockwise from East.
- 3 A hiker walks from point A to point B on a bearing of 070°. Without calculating a distance, explain what the back bearing from B to A must represent and find its value.