C5.2; E5.2 - Three-figure bearings

C5.2; E5.2 - Three-figure bearings

You travel from a harbour on a bearing of 065∘065^\circ. Which bearing takes you directly back? C5.2 and E5.2 give directions a precise meaning: measure clockwise from north at the starting point, draw the route, then use parallel north lines to reverse it.

Bearing conventions

“The bearing of BB from AA” tells you to stand at AA and look towards BB. The word from identifies the vertex of the angle.

Three-figure bearing

A bearing is an angle measured clockwise from north at the starting point. Write it with three figures: 037∘037^\circ, not 37∘37^\circ. Due north is 000∘000^\circ (also 360∘360^\circ), due east 090∘090^\circ, due south 180∘180^\circ, and due west 270∘270^\circ.

Draw a north arrow through the starting point. Follow the clockwise turn all the way to the direction line, even when it exceeds 180∘180^\circ.

[DIAGRAM: asset_name: Clockwise bearings from north; asset_slug: bearing_clockwise; caption: A north-east direction has bearing 035 degrees; a north-west direction has bearing 325 degrees.; description: Two exact compass diagrams with north up and east right. The first ray is 35 degrees clockwise from north. The second ray is 35 degrees west of north, with the full clockwise 325 degree arc marked. North, east, south and west are labelled with three-figure bearings.]
A north-east direction has bearing 035 degrees

Worked example 1: east of north

Question. BB lies 35∘35^\circ east of north from AA. Write down the bearing of BB from AA.

Plan. Start at north at AA and turn clockwise towards BB.

The clockwise angle is 35∘35^\circ. Add a leading zero to write three figures.

Answer. 035∘035^\circ.

Check. The direction lies between north 000∘000^\circ and east 090∘090^\circ, as 035∘035^\circ does.

Only the side of north changes next. The short angle is now anticlockwise, so it is not the bearing.

Worked example 2: west of north

Question. CC lies 35∘35^\circ west of north from AA. Calculate the bearing of CC from AA.

Plan. Subtract the short anticlockwise angle from a complete turn.

360∘−35∘=325∘360^\circ-35^\circ=325^\circ

Answer. 325∘325^\circ.

Check. North-west directions have bearings between west 270∘270^\circ and north 360∘360^\circ. Also 325∘+35∘=360∘325^\circ+35^\circ=360^\circ.

A bearing measures direction, not distance. Moving further along the same ray leaves the bearing unchanged.

Watch out

The smaller angle between north and the route is not always the bearing. Use the clockwise angle, and keep the three figures.

Choose the correct start point and turn in this new case.

Drawing bearings

A bearing fixes a ray; a distance fixes the endpoint on that ray. Use a ruler for straight edges and a protractor for the angle. If the distance is too large for the page, choose or use the given scale.

You already know

At a scale of 1 cm representing 2 km, divide a real distance in kilometres by 2 to get its drawing length in centimetres. A protractor's centre belongs exactly on the starting point.

Keep the north line and the route ray distinct. For a bearing greater than 180∘180^\circ, a semicircular protractor can mark the remaining angle on the west side of north, but the labelled bearing is still the full clockwise turn.

Worked example 3: drawing a north-east route

Question. BB is 8 km from AA on a bearing of 060∘060^\circ. Describe an accurate construction using 1 cm to represent 2 km.

Plan. Draw north at AA, measure the direction, then scale the distance.

  1. Mark AA and draw a vertical north arrow through it.
  2. Centre the protractor on AA, with zero aligned to north. Mark 60∘60^\circ clockwise, on the east side, and draw the ray using a ruler.
  3. Calculate 8÷2=48\div2=4. Measure 4 cm from AA along the ray and label the endpoint BB.

Answer. A 4 cm segment from AA, 60∘60^\circ clockwise from north, with the bearing labelled 060∘060^\circ.

Check. Measuring back gives a 60∘60^\circ clockwise angle; 4×2=84\times2=8 km checks the distance. BB lies north-east of AA.

The next route has the same scale and distance but is west of north.

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