Bearings and 3D problems

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Worksheet with answersDownload PDF

What this lesson covers

  • A bearing is a three-digit angle clockwise from north.
  • Draw a north line at every point.
  • A 3D problem splits into flat triangles.
Key idea

Reading a bearing

  • Measure clockwise from north, and write three digits, such as 060°.
  • Draw a north line at each point before marking angles.
Worked example

A ship's journey

  • A ship sails 20 km on 060°, then 30 km on 150°. The 90° change of course gives a right angle.
  1. d = √(20² + 30²) = 36.06 km
  2. angle at start = tan⁻¹(30/20) = 56.3°
  3. bearing = 60° + 56.3°
  4. bearing = 116.3°
Worked example

A tower in 3D

  • A 40 m tower is seen from A at 30° elevation and from B at 45°. The two lines to its base meet at 90°. Find AB.
  1. base to A = 40 ÷ tan 30° = 69.28 m
  2. base to B = 40 ÷ tan 45° = 40 m
  3. AB² = 4800 + 1600
  4. AB = 80 m

Common mistakes

  • Measuring a bearing from east.
    Bearings always start from north.
  • Treating a slanted line as flat.
    Find the flat distances on the ground first.

Your turn

  • 1. A boat sails 12 km on 090°, then 5 km on 180°. How far is it from the start?
  • 2. What is its bearing from the start?
  • 3. A flagpole is 30 m away at 25° elevation. How tall is it?
Show answer

1. 13 km 2. 90° + 22.6° = 112.6° 3. 30 tan 25° = 13.99 m

Summary

What to take away

  • Draw it first, with a north line at each point.
  • Split 3D shapes into flat triangles.

More in AS 91259

The sine and cosine rulesNotesArea of any triangleNotesBearings and 3D problemsNotesRadians, arcs and sectorsNotes

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