0580

Bearings, Constructions and Scale Drawings

Geometry

The three rules

  • A bearing describes a direction, and it follows three rules without exception
    • Measured from north
    • Measured clockwise
    • Written with three figures, so 59° is recorded as 059°
  • Bearings therefore run from 000° round to 360°
  • The compass directions have fixed bearings: north 000°, east 090°, south 180°, west 270°, with north-east 045°, south-east 135°, south-west 225° and north-west 315°
A compass rose with each of the eight directions labelled with its three-figure bearing: north 000 or 360, north-east 045, east 090, south-east 135, south 180, south-west 225, west 270 and north-west 315, read clockwise from north
Source: Bearings by Save My Exams
  • "Due east" means exactly 090°, and the word "due" signals an exact compass direction

Which point to start from

  • "The bearing of B from A" means stand at A and look towards B, so the north line is drawn at A
  • The point after "from" is where the north line goes, and getting this the wrong way round reverses the answer
  • To measure one, start by drawing north at wherever you are standing, join the two points, and take the clockwise angle with a protractor
  • To plot one, draw the north line, measure the angle clockwise, rule a long line, then mark the distance along it using the scale
A scale drawing at 1 cm to 10 km with its own north arrow: a red north line has been drawn at the point P and a long red line ruled away from P at the measured clockwise angle from that north line
Source: Bearings by Save My Exams

Back bearings

  • The bearing of A from B and the bearing of B from A always differ by exactly 180°
  • Reverse a bearing below 180° by adding on a further 180°
  • Reverse one above 180° by taking 180° away instead, which keeps the answer inside the 000° to 360° range
  • The rule follows from the two north lines being parallel, so the two bearings are co-interior angles
A two-leg journey drawn to scale: a ship sails from P along one bearing to Q, then a fresh north line is drawn at Q and the second leg is ruled from there to the final position, each north line drawn at the point being set off from
Source: Bearings by Save My Exams
Exam tip

Bearings appear in 19 of the 43 papers and are always three figures, so 72° is written 072°. Measure clockwise from the north line at the point named after the word "from": the bearing of B from A is measured at A, and drawing that north line in first is what stops the two ends being swapped.

Worked example

Reversing a bearing both ways

The bearing of B from A is 072°. The bearing of Q from P is 310°. Find the bearing of A from B, and the bearing of P from Q.

Points A and B joined by a straight line, with a dashed north arrow drawn at each of them so the two north lines are parallel. The angle at A, measured clockwise from its north line round to the line AB, is marked 072 degrees. The angle at B is not marked. The figure is marked NOT TO SCALE.

Solution:

  • For the first, 072° is less than 180°, so add 180°
  • 72 + 180 = 252, so the bearing of A from B is 252°
  • For the second, 310° is more than 180°, so subtract 180°
  • 310 − 180 = 130, so the bearing of P from Q is 130°
  • Both answers are written with three figures and lie between 000° and 360°
  • A question may give one bearing as an expression and the other as a number, which turns the same rule into an equation to solve
  • Harder bearings questions combine this with Pythagoras or trigonometry, so draw and label a clear diagram before calculating