An angle measured clockwise from north
The eight-point compass names eight directions, 45° apart. Most directions fall between them, and to name those exactly we use an angle.
A bearing gives the direction from one point to another as an angle, measured from north and turning clockwise. To find the bearing of B from A, stand at A and face north. Turn clockwise until you face B. The angle you have turned through is the bearing of B from A.
In the figure, the turn from north to the line toward B is 115°, so B is on a bearing of 115° from A. That is 25° past east, which is 090°, and short of southeast, which is 135°, so B lies between east and southeast.
The arc turns clockwise from north to the line toward B: a bearing of 115°.
Why a bearing needs both parts
An angle of 115° on its own does not fix a direction, because it does not say where the turn starts or which way it goes. Turn 115° clockwise from north and you face between east and southeast. Turn 115° counterclockwise from north and you face between southwest and west. Turn 115° clockwise from east and you face between south and southwest. That is three different directions from one number.
A bearing always starts from north and always turns clockwise, so each angle names exactly one direction. It also means every bearing is at least 000° and less than 360°, since a turn of 360° faces north again.
Past south, keep counting clockwise. A line pointing between southwest and west is on a bearing of 250°: 180° to reach south, then 70° more. The same line is 110° from north the other way round, but it is never called 110°, because the bearing 110° points between east and southeast.
A bearing of 250°. The arc turns clockwise from north through east and south, 250° in all.
Always three figures
A bearing is always written with three figures. A turn of 47° is written 047°, a turn of 8° is written 008°, and a turn of 250° is already three figures, 250°.
The zeros go in front, where they do not change the number: 047 is still forty-seven. A zero added on the end would make it 470, a different number. Since every bearing is less than 360°, three figures are always enough, and every bearing is written the same way.
A turn of 47° clockwise from north is written as the bearing 047°.
The compass points as bearings
Each point of the eight-point compass is 45° clockwise from the one before it, so the eight points have these bearings: north 000°, northeast 045°, east 090°, southeast 135°, south 180°, southwest 225°, west 270° and northwest 315°.
A direction exactly on one of these is often called due east, due south and so on. North is written 000°, not 360°: a full turn brings you back to where you started.
Due east is a quarter turn clockwise from north: 090°.
Drawing and reading a bearing
To draw a bearing of 070° from a point A, draw a line straight up from A for north. Place the center of the protractor on A with its zero line along the north line, count 70° clockwise, and mark the point. The line from A through that mark is on a bearing of 070°.
A protractor that is a half circle only reaches 180°. For a bearing over 180°, such as 250°, draw the line from A straight down as well, pointing south. South is 180°, so measure 250 − 180 = 70° clockwise from the south line.
To read a bearing, draw the north line at the starting point and measure the clockwise angle from it to the line toward the other point. If the line points to the west side of the north–south line, the bearing is more than 180°: measure the angle clockwise from south and add 180°.
the bearing of B from A is 060°, clockwise from north; the bearing of A from B is 060° + 180° = 240°, because the two north arrows are parallel and the angles at A and B are co-interior
Drag B to a bearing of 250° from A and read both bearings
Drag B round A. The gold arc is the turn clockwise from north at A, and the top line gives the bearing of B from A in three figures. Drag B to 250°: the arc passes east and south on the way. The green line and arc give the bearing back, from B to A.
The usual mistakes
Dropping the leading zero. A turn of 47° from north is the bearing 047°, not 47°.
Measuring counterclockwise. The line 47° clockwise from north is 360 − 47 = 313° from north the other way, but its bearing is 047°. In the same way, the bearing of west is 270°, not 090°.
Measuring from the nearest line instead of from north. A line 20° clockwise past east is on a bearing of 090 + 20 = 110°, not 020°.
Worked example: Turning a Ferry onto a New Course, Clockwise or Counterclockwise
Question A ferry is steering a bearing of 340°. The harbor entrance lies north-east of the ferry, and the captain turns onto that course. (a) What is the new course as a three-figure bearing, and through what angle must the ferry turn clockwise to reach it? (b) The captain could turn counterclockwise instead. Which turn is shorter, and by how many degrees?
1.North-east is one step of 45° clockwise from north, so (a) the new course is a bearing of 045°.
The ferry is steering 340°, and the harbor lies north-east of it. 2.Turning clockwise takes the ferry from 340° past north and on to 045°. Because 045 is the smaller number, add a full turn before subtracting: the clockwise turn is 045 + 360 − 340 = 65°.
(a) North-east is one step of 45° clockwise from north, so the new course is 045°. 3.The two turns together make one full turn, so the counterclockwise turn is 360 − 65 = 295°.
Turning clockwise passes north, so add a full turn before subtracting: 045 + 360 − 340 = 65°. 4.(b) The clockwise turn of 65° is the shorter one, and it is shorter by 295 − 65 = 230°. Check: 65 + 295 = 360°, one full turn, as it must be.
The other way round is 360 − 65 = 295°, so (b) the clockwise turn is shorter, by 295 − 65 = 230°.
Answer: (a) 045°, a turn of 65° clockwise; (b) the clockwise turn is shorter, by 230°, since the counterclockwise turn is 295°
Common mistakes
- Working out the turn as 340 − 45 = 295° and calling that the clockwise turn. Subtracting that way measures the turn counterclockwise; the clockwise turn from 340° passes north and is the 65° left over.
- Giving the answer to part (b) as 295°, the size of the longer turn, when the question asks how much shorter the shorter turn is. That is the difference between the two turns, 295 − 65.
More compass directions and bearings problems, worked step by step →
The bearing back
Part (b) of the next application asks for the direction from the vessel back to the station. That direction is exactly opposite, and facing the opposite way is a half turn, 180°. So the bearing back is the bearing out with 180° added. If the bearing out is more than 180°, take 180° away instead, so that the answer stays under 360°. The two always differ by exactly 180° because the north lines at the two ends of the journey are parallel.
Worked example: A Coastguard Log Written Without the Leading Zero
Question A coastguard's log must record every bearing with three figures. One morning a trainee writes two entries as "8" and "80". The first vessel lies 8 degrees clockwise from north of the station, and the second lies 80 degrees clockwise from north of the station. (a) Write each of the two bearings correctly, with three figures. (b) A lifeboat is sent from the station to the first vessel. What is the bearing of the station from that vessel?
1.A bearing runs from 000° to 360°, so three figures are always enough and always used. A bearing under 100° needs one leading zero and a bearing under 10° needs two.
A bearing runs from 000° to 360°, so it is always written with three figures. 2.The first vessel lies 8 degrees clockwise from north, so (a) its bearing is written 008°.
(a) The first vessel lies 8 degrees clockwise from north, a bearing of 008°. 3.The second vessel lies 80 degrees clockwise from north, so its bearing is written 080°. Written as "80" it could be read as 800, which is not a bearing at all, because a bearing is never 360° or more.
The second lies 80 degrees clockwise from north, a bearing of 080°; written as 80 it could be read as 800. 4.The bearing of the first vessel from the station is 008°. The bearing of the station from that vessel is a half turn from it, and 008° is less than 180°, so add the half turn: (b) 008 + 180 = 188°.
(b) The bearing of the station from A is 008 + 180 = 188°. 5.Check: 188 − 180 = 008, the bearing the lifeboat went out on, and 188° is just past due south, which is where the station must be from a vessel almost due north of it.
Check: 188 − 180 = 008, the bearing the lifeboat went out on.
Answer: (a) 008° and 080°; (b) 188°
Common mistakes
- Writing the first bearing as 800°, filling the three figures with a zero on the end instead of the front. A zero on the end multiplies the angle by ten; a zero on the front leaves it unchanged.
- Giving the bearing in part (b) as 180 − 008 = 172°. The half turn is added to the bearing out, not taken from 180°, because the two north lines are parallel and the directions are opposite.
More compass directions and bearings problems, worked step by step →